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Bookseller  aod  SlatioDer, 

AND  riEALKK  IN 

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ELEMENTS 


OF 


GEOMETRY    AND    TRIGONOMETRY, 


I 


PnOM      THE      WOUKS      OP 


A.    M.    LEGENDRE. 


ADAPTED   TO   THE   COUKSE   OF   MATHEMATICAL    INSTRUariON    tN 

THE   UNITED    STATES, 


BY     CHARLES    DAVIES,    LL.D., 

iJTHOR     OF     ARITHMETIC,     ALGEBRA,      PRACTICAL    MATHEMATICS    FOR    PRACTICAL    MSN, 

ELEMENTS    OF    DESCRIPTIVE    AND    OF    ANALYTICAL    GEOMETRY,    ELEMENTS 

OF    DIFFERENTIAL    AND    INTEGRAL    CALCULUS,    AND    SHADES, 

SHADOWS,    AND    PERSPECTIVE. 


A.   S.  BARNES    &  COMPANY, 
NEW  YORK  AND  CHICAGO. 

1874. 


DAVIES'  MATHEMATICS.  ^S^S. 


»♦-*- 


And  Oril>  Thorough  and  Complete  Mathematical  Series. 


'®  J<-' 


I3>3"     TI€:K,EE     I3.A.S.TS. 


/.    COMMON   SCHOOL    COURSE 

Davies'  Primary  Arithmetic  — The  fuudamental  principles  displayed  in 

Object  Lessons. 
Savies'  Sntellectual  Arithmetico— Referring  all  operations  to  the  unit  1  as 

the  only  tangible  basis  for  logical  development.  . 
Davies'  Elements  of  Written  Arithmetic— A  practical  introduction  to 

the  whole  subject.    Theory  subordinated  to  Practice. 
SSavies"  Practical  Arithmetic.*— The  most  successful  combination  of  Theory 

and  Practice,  clear,  exact,  brief,  and  comprehensive. 

//.  ACADEMIC  COURSE. 

Davies'  University  Arithmeticn*— Treating  the   subject  exhaustively  as 

a  science,  in  a  logical  series  of  connected  propositions. 
Davies''  Sllementary  Algebra.*— A  connecting  link,  conducting  the  pupil 

easily  from  arithmetical  processes  to  abstract  analysis. 
Davies'  University  Algebra.*- For  institutions  desiring  a  more  complete 

but  not  the  fullest  course  in  pure  iMgebra. 
Davies'  Practical  BSathematics.— The  science  practically  applied  to  the 

useful  arts,  as  Drawing,  Architecture,  Surveying,  Mechanics,  etc. 
Davies'  Elementary  Geometry.- The  important  principles  in  simple  form, 

but  with  all  the  exactness  of  vigorous  reasoning. 
Davies'  Slleinents  of  Surveying.— Re-written  in  18~0.     The  simplest  and 

most  practical  presentation  for  youths  of  12  to  16. 

///.  COLLEGIATE  COURSE. 

Davies'  Bourdon's  Algebra.*— Embracing  Sturm's    Theorem,  and  a  most 
exhaustive  and  scholarly  course. 

Davies'  University  Algebra.*- A  shorter  course  than  Bourdon,  for  Institu- 
tions have  less  time  to  give  the  subject. 

Davies'  Legendre's  Geometry.— Aclmowledged  ^Aeo;i^y  satisfactory  treatise 
of  its  grade.    300,000  copies  have  been  sold. 

Davies'  Analytical  Geometry  and  Calculus.— The  shorter  treatises, 
combined  in  one  volume,  are  more  available  for  American  courses  of  study. 

Savies'  Analytical  Geometry.  I  The  original  compendiums,  for  those  de- 

i)avies'  SilT.  &  Int.  Calculus.     '     siring  to  give  full  time  to  each  branch. 

Davies'  Descriptive  Geometry o-'^ith  application  to  Spherical  Trigonome- 
try, Spherical  Projections,  and  ^^'arped  Surfaces. 

Davies'  Shades,  Shadows,  and  Perspective.-A  succinct  exposition  of 
the  mathematical  principles  involved. 

Davies'  Science  of  Mathematics.— For  teachers,  embracing 

I.  Gramsiab  of  Arithmetic,  III.  Logic  and  Utilitt  of  Mathematics, 

n.  Outlines  of  Mathematics,  TV".  Mathematical  Dictionary. 


Keys  may  be  obtained  from  the  Publishers  by  Teachers  only. 


Entered,  according  to  Act  of  Congress,  in  the  year  1S62,  by 

CHARLES     DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southeni  District  of 

New  York. 

L.  GIFT  OF    r  U.S.Wa 


iji, 


TO         EflQINrrrpiw/-  r  .m-. ^ 


PREFACE. 


Of  the  various  Treatises  on  Elementary  Geometry 
which    have    appeared    during    the     present    century,    that  f 

of  M.  Legendke  stands  preeminent.  Its  peculiar  merits 
have  won  for  it  not  only  a  European  reputation,  hut 
have  also  caused  it  to  be  selected  as  the  basis  of  many 
of  the  best  works  on  the  subject  that  have  been  pub- 
lished  in   this    country. 

In  the  original  Treatise  of  Legendre,  the  propositions 
are  not  enunciated  in  general  terms,  but  by  means  of 
the  diagrams  employed  in  their  demonstration.  This 
departure  from  the  method  of  Euclid  is  much  to  be 
regretted.  The  propositions  of  Geometry  are  general 
truths,  and  ought  to  be  stated  in  general  terms,  without 
reference  to  particular  diagrams.  In  the  following  work, 
each  proposition  is  lirst  enunciated  in  general  terms,  and 
afterwards,  with  reference  to  a  particular  figure,  that 
figure  being  taken  to  represent  any  one  of  the  class  to 
which  it  belongs.  By  this  arrangement,  the  difficulty 
exper.cnced  by  beginners  in  comprehending  abstract  truths, 
is  lessened,  without  in  any  manner  impairing  the  gener- 
ality of  the   truths   evolved. 

The  term  solid,  used  not  only  by  Legendre,  but  by 
many  other  authors,  to  denote  a  limited  portion  of  space, 
eeeras   calculated    to  iii^^^i^p^/fjW   foreign    idea   of  matter 


iv  PREFACE. 

into  a  science,  whicli  deals  only  with  the  abstract  pro- 
perties and  relations  of  figured  space.  The  term  volume^ 
has  been  introduced  in  its  place,  under  the  belief  that 
it  corresponds  more  exactly  to  the  idea  intended.  Many 
other  departures  have  been  made  from  the  original  text, 
the  value  and  utility  of  which  have  been  made  manifest 
in  the  practical  tests  to  which  the  work  has  been  sub- 
jected. 

In  the  present  Edition,  numerous  changes  have  been 
made,  both  in  the  Geometry  and  in  the  Trigonometry.  The 
definitions  have  been  carefully  revised — the  demonstrations 
have  been  harmonized,  and,  in  many  instances,  abbreviated — 
the  principal  object  being  to  simplify  the  subject  as  much  aa 
possible,  without  departing  from  the  general  plan.  These 
changes  are  due  to  Professor  Peck,  of  the  Department  of 
Pure  Mathematics  and  Astronomy  in  Columbia  College.  For 
his  aid,  in  giving  to  the  work  its  present  permanent  form,  I 
tender  him  my  grateful  acknowledgements. 

CHARLES   DATIES. 

Columbia  CoiiLEoa, 
Nbw  Yoek.,   April,    1862. 


CONTENTS. 


GEOMETRY. 

Fi.as. 

[?rTRODT7CTI0X     9 

BOOK     I. 

Definitions, 13 

Propositions, 20 

BOOK     n. 
Ratios  and   Proportions, 50 

BOOK      HL 

The   Circle,  and   the  Measurement   of  Angles, 69 

Problemfl  relating   to   the  First  and   Third  Books, 82 

BOOK    rv. 

Proportions   of  Figures — Measurement  of  Areas, 03 

Problems   relating   to   the   Fourth   Book, 129 

BOOK      V. 
Regular  Polygons — Measurement  of  the  Circle, 136 

BOOK     VI. 
Planes,   and   Polyedral   Angles, 167 

BOOK    vn. 

PolyedroM, ,.,.... , , . , , 178 


VI  CONTENTS. 

BOOK    Yni. 

PAGE. 

Cyliudcr,    Couc,    aud    Sphere, 210 

BOOK     IX. 
Spherical    Geometry, 2S5 

PLANE     TRIGONOMETRY. 

IXTllODUCTION. 

Definition  of  Logarithms, 3 

Rules   for   Ciiaracteristics,     -t 

General  Principles,     6 

Table  of  Logarithms,    7 

Manner  of  Using   the   Table,    8 

Multiplication    by  Logarithms, 11 

Division    by   Logarithms, 12 

Arithmetical    Complement,   1^ 

Raising   to  Powers  by  Logarithms, 15 

Extraction   of  Roots   by  Logarithms,   16 

PLANE     TRIGONOMETRY. 

Plane   Trigon'jmetry  DeGneJ 17 

Functions   of  the  Arc, 18-22 

Table  of  Natural  Sines, 22 

Table   of  Logarithmic   Sines,     22 

Use   of  the   Table, 23-27 

Solution  of  Right-angleJ  Triangles, 27-35 

Solution   of  Oblique-angled    Triangles,     dC-J7 

Problems  of  Application, 48 

ANALYTICAL     TRIGONOMETRY. 

Analytical   Trigonometry   Defined, 51 

Definitions   and  General   Principles, . .    5 1--54 

Rules  for  Signs   of  the   Functions,. 61 


CONTENTS.  vh 

PAGE. 

fjnilling    vu'.iie   of  Circular    Functions,   55 

Relations   of  Circular   Functions,      57-59 

Functions   of  Negative    Arcs,    60-C2 

Particular   values  of  Certain    Functions,    C3 

Formulas   of  Relation    between    Functions  and  Arcs, 64--G6 

Functioiib   of   Double    and    Half  Arcs,    (i7 

Additional   Formulas, C8-7f 

Method   of  Computing   a   Table   of  Natural  Sines, 71 

SPHERICAL     TRIGONOMETRY. 

Spherical  Trigonometry   Defined,    73 

General    Friiiciples, 73 

Formulas    for   Right-angled   Triangles,     74-76 

Napier's  Circular   Parts,     77 

Solution    of  Right-angled   Spherical   Triangles, 80-83 

Quadrantal    Triangles, 84 

Formuliis   for   Oblique-angled  Triangles, 85-92 

Solution   of  Oblique-angled   Triangles, 92-  104 

MENSURATION. 

Mensuration  Defined, 1 05 

The   Area  of  a   Parallelogram, 106 

The    Area   of  a   Triangle, 106 

Formula  for  the  Sine   of  Half  an  Angle, 108 

Area  of  a   Trapezoid,   112 

Area   of  a   Quadrilateral,   112 

Area    of  a  Polygon,    1 11} 

Area  of  a   Regular   Polygon,   11  -1 

To   find    the    Circumference    of  a  Circle,    ll(i 

To   find   tlie    Diameter  of  a   Circle,     116 

To   find   the   length    of  an    Arc, 117 

Area   of  a    Circle 117 

Area   of  a  Sector, ■. 118 

Area   of  a   Segment, 118 

Area   of  a   Circular  Ring, IIP 


viii  CONTENTS. 

PAGE. 

Area   of  the   Surface   of  a   Prism, 120 

Area  of  the   Surface   of  a   Pyramid, 120 

Area   of  the   Frustum   of   a   Cone,     121 

Area   of  t!ie   Surfoce   of  a   Sphere,     122 

Area   of  a    Zone,     122 

Area   of  a   Spherical   Polygon,    123 

V>lume   of  a    Prism,   12-i 

Volume   of  a    Pyramid, 124 

Volume   of  tlie    Frustum    of  a   Pyramid, 125 

Volume   of  a   Sphere, • 12C 

Volume   of  a   Wedge,     127 

Volume   of  a   Prismoid, ■ 128 

Volumes  of  Regular   Polyedroag, 1S2 


I  n 


ELEMENTS 


OF 


GEOMETRY 


INTRODUCTION. 

DEFIKITI02q"S     OT"     TEEMS. 

1.  QUANTITY  is  anything  which  can  be  increased,  dimin- 
ished,  and  measured. 

To  measure  a  thing,  is  to  find  out  how  many  times  it 
contains  some  other  thing  of  the  same  kind,  taken  as  a  stand- 
ard.    The  assumed   standard   is   called   the   unit   of  measvre. 

2.  In  Geometry,  there  are  four  species  of  quantity,  viz.: 
LiXES,  SuEFACES,  VOLUMES,  and  Angles.  These  are  called, 
GEOMETracAL  Magnitudes. 

Since  the  unit  of  measure  of  a  quantity  is  of  the  same 
kind  as  the  quantity  measured,  there  are  four  kinds  ot  units 
of  meapure,  viz.:  Units  of  Length,  Units  of  Surface,  Units 
of   Volume,  and    Units  of  Angular  Ifcasure. 

3.  Geometry  is  that  branch  of  Mathematics  which  treats 
of  the  properties,  relations,  and  measurement  of  the  Geo- 
m.etrical   Magnitudes. 

4.  In  Geometry,  the  quantities  considered  are  generally 
represented  by  means  of  the  straight  line  and  curve.  The 
operations  to  be  performed  upon  the  quantities  and  the  rela- 
tions  between  them,   are   indicated   by   signs,   as   in  Analysis. 


10  GEOMETRY. 

TliG  f)l!o^vir,g   aro   tlie   pnncipal   signs   employed  : 

The   Sign   of  Addition,     +  ,     called   ^)/(<s  ; 

Thus,    A  +  JJ,     mtlicatcs   that     H     is   to   be   added   to   A. 

The    Sign   of  Subtraction,     —  ,     called   mi7ius  : 
Tims,    A  —  B,      indicates    that     B     is    to    be    subtracted 
from    A. 

The   Sigti   of  3fuUiplication,     x  : 

Thu3j  A  X  B,  indicates  that  A  is  to  be  multii)lied 
by    B. 

The   Sign   of  BivisioJi,     ~  : 

Thus,  A-j^B,  or,  -^, ,  indicates  that  A  is  to  be 
divided     by     B. 

The   J^'X2?onenticd  Sign  : 

Thus,  A^ ,  indicates  that  A  is  to  be  taken  three  times 
as   a   factor,    or   raised   to   the   tliird   power. 

The   Radical   Sign,     -y/      : 

Thus,  .^fA,  \fB,  indicate  that  the  square  root  of  A^ 
and   the    cube   root   of    B,     are   to   be   taken. 

Yfhen  a  compound  quantity  is  to  be  operated  upon  as  a 
single  quantity,  its  parts  are  connected  by  a  ^^culum  or 
by   a  parenthesis  : 

Thus,  A  ^  B  X  C,  indicates  that  tlie  sum  of  A  and 
B  is  to  be  niultii)lied  by  G ;  and  {A  -\-  B)  -^  C,  indi- 
cates  that    the   sum   of    A     and    B     is   to   be   divided   by    C. 

A  luimber  -written  before  a  quantity,  shows  liow  manv 
times   it   is   to   be   taken. 

Thus,     3(^1  +  B),     indicates   that    the    sum    of    A     and     1 
is  to   be   taken   three   times. 

The  Sign   of  Equality,     =  : 

Thus,  A  =  B  -\-  C,  indicates  that  A  is  equal  to  tbe 
6um   of    B    and     C. 


I N  T  R  O  D  U  C  T  ION.  11 

The  expression,  A  =  Jj  -\-  C,  is  called  nii  equation.  The 
part  on  the  left  of  the  sign  of  equality,  is  called  the  Jirsi 
7n€mber  j   that   on   the   right,  tlie   second  member. 

The    Sign  of  Inequality^     <   : 

Thus,  ^/A  <  \/B,  indicates  tliat  the  square  root  of  A 
is  less  than  the  cube  root  of  B.  The  opening  of  the  sigu 
is  towards  the   greater   quantity. 

The  sigu,  .'.  is  used  as  an  abbreyiation  of  the  word 
hence,   or  consequently. 

The  symbols,   1°,   2°,  etc.,  mean,   1st,  2d,   etc. 

5.  The  general  truths  of  Geometry  are  deduced  by  a 
course  of  logical  reasoning,  the  premises  being  definitions  and 
principles  previously  established.  The  course  of  reasoning 
employed  in  establishing  any  truth  or  principle,  is  called  a 
demonstration. 

6.  A  Theoee^j  is   a   truth   requiring   demonstration. 

7.  An   Axiom   is  a  self-evident  truth. 

8.  A   Peoblem   is   a   question   requiring   a   solution. 

9.  A   Postulate  is  a  self-evident  Problem. 

Theorems,  Axioms,  Problems,  and  Postulates,  are  all  called 
Projjositions. 

10      A   Lemma   is   an   auxiliary  proposition. 

11,  A  Corollary  is  an  obvious  consequence  of  one  or 
move   propositions. 

12.  A  Scholium  is  a  remark  made  upon  one  or  more 
propositions,  with  reference  to  their  connection,  their  use, 
their   extent,   or   their   limitation. 


12  GEOMETRY. 

13.  An  HrroTnESis  is  a  supposition  made,  either  in  the 
statement  of  a  proposition,  o]-  in  the  course  of  a  demonstra- 
tion. 

14.  Magnitudes  are  equal  to  each  other,  when  each  con- 
tiins   the   same   unit   an    equal   number   of  times. 

15.  Magnitudes  are  equal  in  all  their  parts^  when  they 
may  be  so  placed  as  to  coincide  throughout  their  whole 
exteut. 


ELEMENTS    OF    GEOMETRY. 


B  O  O  K    I. 

BLEMKNTART       PRINCIPLES. 
DEFIKLTIONS. 

1.  GE03i[ETRY  is  that  branch  of  Mathematics  which  treats 
of  the  properties,  relations,  and  measurements  of  the  Geo- 
metrical Magnitudes. 

2.  A  Point  is  that  which  has  position,  but  not  magni- 
tude. 

3.  A  Line  is  that  which  has  length,  bu^  neither  breadth 
nor  thickness. 

Lines   are   divided   into   two   classes,   sfraif/Jit   and   curved. 

4.  A  Steaight  Line  is  one  which  does  not  change  its 
jiirection   at   any  point. 

5.  A  CuKYED  Line  is  one  which  changes  its  direction  at 
every  point. 

When  the  sense  is  obvious,  to  avoid  repetition,  the  word 
'ine,  alone,  is  sometimes  used  for  straigM  line;  and  the 
word   curve,   alone,   for   curved  line. 

G.  A  line  made  up  of  straight  lines,  not  lying  in  the  same 
direction,   is   called   a   broken   line. 

7.  A  Surface  is  that  which  has  length  and  breadth 
without   thictness. 


14 


GEOMETRY. 


Surfaces  are  divided  into  two  classes,  ^>/a;ie  and  curved 
surfaces. 

8.  A  Plants  is  a  surface,  such,  that  if  any  two  of  its 
points  be  joined  by  a  straight  line,  that  Une  will  lie  wholly 
in   the    surface. 

9.  A  Curved  Surface  is  a  surface  which  is  neither  u 
plane   nor   composed   of  planes. 

10.  A  Plaxe  Angle  is  the  amonnt  of  divergence  of  two 
straight  lines   lying  in   the   same  ]ilane. 

Thus,  the  amount  of  divergence  of  the 
lines  AJ3  and  AC,  is  an  angle.  The 
lines  AH  and  AC  are  called  sides,  and 
their  common  point  A,  is  called  the  ver- 
tex. An  angle  is  designated  by  naming  its  sides,  or  some- 
times by  simply  naming  its  vertex  ;  thus,  the  above  is  called 
the   angle   BAG,    or  simply,   the   angle    A. 

11.  When      one      straight     line      meets 
another    the     two    angles   which    they    form 
are      called      adjacent     angles.        Thus,     the    -^  i3~ 
angles     AJBD      and      DBC      are    adjacent. 

12.  A  Right  Angle  is  formed  by  one 
straight  line  meeting  another  so  as  to 
make  the  adjacent  angles  equal.  The  first 
line   is  then   said   to   be  perpendicular  to   the   second. 

13.  An  Oblique  Angle  is  formed  by 
one  straight  line  meeting  another  so  as 
to   make   the   adjacent   angles   unequal.  ' 

Oblique  angles  are  subdiWdcd  into  two  classes,  acut^ 
angles,   and   obtuse   angles. 

14.  An  Acute  Angle  is  less  than  a 
right  angle 


B  O  0  K    1 .  15 


15.     An   OuTUgE  A>fGLE   is   greater  than 
a   riiilit   ansjle. 


16.  Two  straight  lines  are  parallel, 
when  they  lie  in  tlie  same  plane  and  can- 
not  meet,  how  far  soever,  either  way,  both 

may  be   i:)roduce(l.      They  then   have   the   same  direction. 

17.  A  Plane  Figuke  is  a  portion  of  a  plane  bounded 
by   lines,    cither   straight   or   curved. 

18.  A  Polygon  is  a  plane  figure  bounded  by  straight 
lines. 

The  bounding  lines  are  called  sides  of  tlie  polygon.  The 
broken  line,  made  up  of  all  the  sides  of  the  polygon,  is  called 
the  perimeter  of  the  polygon.  The  angles  formed  by  the 
sides,    are    called   angles   of  the   polygon. 

10.  Polygons  are  classified  according  to  the  number  of 
their   sides   or   angles. 

A  Polygon  of  three  sides  is  called  a  trianr/le ;  one  of 
four  sides,  a  quadrilateral ;  one  of  five  sides,  a  lyentagon  ; 
one  of  six  sides,  a  hexagon ;  one  of  seven  sides,  a  heptor 
gon  /  one  of  eight  sides,  an  octagon  /  one  of  ten  sides,  a 
decagon  ;    one   of  twelve   sides,   a  dodecagon,   &c. 

20.  An  Equilateral  Polygon,  is  one  whose  sides  are 
all    equal. 

An  Equiangular  Polygon,  is  one  Avhose  angles  are  all 
equal. 

A  Regular  Polygon,  is  one  which  is  both  equilateral 
and    equiangular. 

21.  Two  polygons  are  vndualhj  equilateral,  when  their 
sides,  taken  in  the  same  order,  are  equal,  each  to  each :  that 
is,   following   their  perimeters   in    tlie   same   direction,  the  first 


16  GEOMETRY. 

side  of  the  one  is  equal  to  the  first  side  of  tlie  other,  the 
second  side  of  the  one,  to  the  second  side  of  the  other, 
und   so   on. 

22.  Two  polygons  are  mutually  equiangular,  when 
their  angles,  taken  in  the  same  order,  are  equal,  eaoli  to 
each. 

23.  A  Diagonal  of  a  polygon  is  a  straight  line  joiniiig 
the  vertices   of  two   angles,   not  consecutive. 

24.  A   Base   of    a   polygon     is    any    one    of    its    sides    ou 
which  the    polygon    is    supposed    to    stand. 

25.  Triangles  may  be  classiiied  with  reference  either  to 
their   sides,   or   their   angles. 

When  classified  with  reference  to  their  sides,  there  are 
two    classes  :    acalene   and    isosceles. 

1st.     A  Scalene  Triangle  is  one  which 
has   no   two   of  its   sides   equal, 

2d.     An  Isosceles  Tutangle  is  one  which 
has   two   of  its   sides   equal. 

When   all    of    the    sides   are    equal,    the 
triangle   is   equilatesal. 

When  classified  with  reference  to  their  angles,  there  are 
two   classes  :   right-a^igled   and   oblique-angled. 

1st.     A  Right-angled  Triangle  is   one 
that   has   one   right   angle. 

The  side  opposite  the  right  angle,  is  called  the  li.ypotlw 
iiuse. 

2d.     An     Oblique-angled    Triangle    is 
owe   whose   angles   are   all   oblique. 


BOOK    I. 


17 


If  one  angle  of  an  oblique-angled  triangle  is  obtuse,  the 
triangle  is  said  to  be  obtuse-ajstgled.  If  all  of  the  angles 
are  acute,  the  triangle  is  said  to  be  acute-ai^gled. 

26.     Quadrilaterals   are   classified  with  reference  to  the   rel- 
ative  directions   of  their  sides.      There   are  then  two   classes 
the  first  class   embraces  those   which  have  no   two   sides  par 
aliel;    ilie  second  class    embraces  those  which    have    at    least 
two  sides  parallel. 

Quadrilaterals  of  the  first   class,   are   called   tra2->ezmms. 

Quadrilaterals  of  the  second  class,  are  divided  into  two 
species  :   trapezoids  and  parallelograms. 


27.     A    Trapezoid    is     a     quadrilateral 
which   has   only   two    of  its   sides  parallel. 


28.     A  Paeallelogram    is    a    quadrilateral    which    has    its 
opposite   sides  parallel,   two   and  two. 

There     arc    two    varieties    of    paraUelograma  :     rectangles 
and  rhomboids. 

1st.     A    Rectanglb    is    a    parallelogram 
whose   angles   are  aU  right  angles. 


A    Square    is  an    equilateral    rectangle. 


2d.    A    RnoMBom    is    a    parallelogram 
whose   angles   are   all   oblique. 


A  Rno:iiBus  is  an  equilateral  rhomboid. 


/ 

/ 

/ 

7 

I 

'  / 

18  GEO  METE  Y. 

29.  Space  is  indefinite  extension. 

30.  A  Volume  is   a   limited  portion   of  space,   combining 
the  three  dimensions  of  length,  breadth,  and  thickness. 

AXIOMS. 

1.  Thmgs  which   are   equal  to    the   same  thing,   are   equa 
t.)   each   other. 

2.  If  equals  be   added  to   equals,  the   suras  will  be  equal. 

3      If   equals    be    subtracted   from    equals,   the    remainders 
will  be   equal. 

4.  If    equals    be    added    to    unequals,    the    sums    wiU    be 
unequal. 

5.  If  equals  be   subtracted   fi'om   unequals,   the   remainders 
will  be   unequal. 

6.  If  equals  be  multipUed  by  equals,  the  products  wiU  be 
equal. 

7.  If   equals  be   divided  by   equals,   the   quotients   will    be 
equal. 

8.  The   whole  is  greater  than   any  of  its  parts. 

9.  The   whole  is   equal  to  the   sum   of  all  its  jiarts. 
10.     All   right  angles   are   equal. 

11.  Only   one    straight    line    can    be    drawn   joining    two 
given  points. 

12.  The  shortest  distance    from  one    point    to  another  is 
measured  on  the  straight  line  which  joins  them. 

13.  Through  the  same  point,  only  ©ne  straight  lino  can 
be  drawn  parallel  to  a  given  straight  line. 


BOOK    I.  19 

POSTULATES. 

1.    A  straight  line  can  be  drawn  joining  any  two  points. 

.    2.    A  straight  line   may  be  prolonged  to   any  length. 

S      If   two  straight    lines  are   uneqnal,  the    length  of    the 
less  may  be  laid  off  on  the  greater. 

4.  A  straight  line  may  be  bisected;   that  is,  divided  into 
two   equal  parts. 

5.  An   angle  may  be  bisected. 

6.  A  perpendicular  may  be  drawn  to  a  given  straight  line, 
either  from  a  point  without,  or  from  a  point  on  the  line. 

7.  A  straight  line  may  be  drawn,  making  with  a  given 
straight  line  an   angle  equal  to   a  given  angle. 

8.  A  straight  line  may  be  drawn  through  a  given  point, 
parallel  to  a  given  line. 

NOTE.. 

In  making  references,  the  following  abbreviations  are  employed,  viz. : 
A.  for  Axiom ;  B.  for  Book  ;  C.  for  Corollary  ;  D.  for  Definition  ;  I.  for 
Introduction  ;  P.  for  Proposition  ;  Prob.  for  Problem  ;  Post,  for  Postu- 
late ;  and  S.  for  Scholium.  In  referring  to  the  same  Book  the  number 
of  the  Book  is  not  given ;  in  referring  to  any  other  Book,  the  number 
of  the  Book  is  given. 


20 


GEOMETRY. 


prwOPOSiTioisr   i.     theoeem. 


If  a  straight  line  meet  another  straight  line^  the  sum  of  the 

adjacent  angles  will  he  equal  to  two  right  angles. 

Let  DG  meet  AB  at  (7 : 
then  will  the  sum  of  the  angles 
DC  A  and  DCB  be  equal  to 
two   right   angles. 

A\  (7,  let  GE  be  drav\n  per- 
pendicular to  AB  (Post.  6)  ;  then, 
by    definition    (D.   ]2),    the    angles 

EGA     and     EGB      will     both    be    right    angles,    and    conse- 
quently,   their   sum   will   be   equal  to    two   right   angles. 

The  angle  DGA  is  equal  to  the  sum  of  the  angles 
EGA    and    EGB     (A.  9)  ;    hence, 

BGA  +  BGB  =  EGA  +  EGB  +  BGB  ; 
But,  EGB  ^  BGB    is   equal   to    EGB     (A.  9);      hence, 

BGA  +  BGB  =  EGA  +  EGB. 

The  sum  of  the  angles  EGA  and  EGB,  is  equal  to 
two  right  angles ;  consequently,  its  equal,  that  is,  the  sum 
of  the  angles  BGA  and  BGB,  must  also  be  equal  to  two 
right   angles  ;    ichich  was   to  be  proved. 

Cor.  1.  If  one  of  the  angles  BGA,  BGB,  is  a  right 
angle,   the   other  must   also   be   a  right   angle. 

Cor.  2.  The  sum  of  the  an- 
gles BAG,  GAB,  BAE,  EAF, 
formed  about  a  given  point  on 
the  same  side  of  a  straight  line 
BF,  is  equal  to  two  right  an- 
gles.      For,  their  sum  is   equal  to 


BOOK    I. 


21 


the  sum  ot    the  angles  EAB    and    EAF\    which,   from  the 
proposition  just   demonstrated,  is   equal  to  two   right  angles. 


DEFijsrrnoNS. 

If  two  straight  lines  intersect  each  other,  they  form  four 
angles  about  the  point  of  intersection,  which  have  received 
different  names,   with  respect  to   each   other. 

1°.  Adjacent  Angles  are 
those  which  lie  on  the  same  side 
of  one  line,  and  on  opposite  sides 
of  the  other ;  thus,  A  GE  and 
EGB,  or  AGE  and  AGD,  are 
adjacent   angles. 

2''.  Opposite,  or  Vektical  Angles,  are  those  which  lie 
on  opposite  sides  of  Loth  hnes ;  thus,  A  GE  and  J)  GB^ 
or  AGD  and  EGB^  are  opposite  angles.  From  the  pro- 
position just  demonstrated,  the  sum  of  any  two  adjacent 
angles  is   equal  to   two   right   angles. 


PEOPOSITION     n.      THEOREM. 


If    two    Straight    lines    mtersect    each  other^    the    opposite    or 
vertical  angles  will  he  equal. 

Let  AB  and  BE  intersect 
at  G  :  then  will  the  opposite 
or   vertical   angles  be   equal. 

The  sum  of  the  adjacent  angles 
AGE  and  AGD,  is  equal  to 
two  right  angles  (P.  I.)  :   the  sum 

of  the  adjacent  angles  AGE  and  EGB,  is  also  equal  tc 
two  right  angles.  But  things  which  are  equal  to  the  same 
thing,   are   equal  to   each   other     (A.   1)  ;    hence, 


22 


GEOMETRY. 
ACE-\-  ACB  =  ACE+  ECB  \ 


Taking  from  both  the  common 
angle  ACE  (A.  3),  there  re- 
mains, 

ACB  =  EGB. 

In   like  manner,  we  find, 

ACD  +  ACE  =  ACD  +  DCB  ; 
and,  taking  away  the  common  angle    ACD,    we  have, 

ACE  =  BCB. 

Hence,   the  proposition  is  proved. 

Cor.  1.  If  one  of  the  angles  about  (7  is  a  right  angle, 
all  of  the  others  will  be  right  angles  also.  For,  (P.  I.,  C.  1), 
each  of  its  adjacent  angles  will 
be  a  right  angle ;  and  from  the 
proposition  just  demonstrated,  its 
opposite  angle  will  also  be  a  right 


D 


angle. 


C 


-B 


Cor.   2.     If   one    Ime    BE,    is  ^' 

perpendicular  to  another  AB,  then  will  the  second  line  AB 
be  perpendicular  to  the  first  BE.  For,  the  angles  BCA 
and  BCB  are  right  angles,  by  definition  (D.  12)  ;  and 
from  what  has  just  been  proved,  the  angles  ACE  and 
BCE  are  also  right  angles.  Hence,  the  two  lines  are 
mutually  perpendicular  to   each   other. 

Cor.  3.  The  sum  of  all  the 
angles  ACB,  BCB,  BCE,  ECF, 
FCA,  that  can  be  formed  about 
a  point,  is  equal  to  four  right 
angles. 


BOOK    I.  23 

For,  if  two  lines  bo  drawn  through  the  point,  mutually 
perpendicular  to  each  other,  the  sum  of  the  angles  which 
they  form  Avill  be  equal  to  four  right  angles,  and  it  will 
also  be  equal  to  the  sum  of  the  given  angles  (A.  9).  Ilencc, 
the   sum  of  the  given   angles  is   equal  to   four  riglit    angles. 


PEOPOSITION     in.       THEOEEM. 

If  two  straight  lines  have  two  points  in  common^  they  vrlll 
coincide  throughout  their  whole  extent,  and  form  one  and 
the  same  line. 

Let  A    and  3     be  two  points 
common   to   two    lines  :     then   will        ^- 
the   lines   coincide   throughout. 

Between  A  and  B  they  must 
coincide  (A.  11),  Suppose,  now,  that  they  begin  to  separate 
at  some  point  (7,  beyond  AB,  the  one  becoming  ACE^ 
and  the  other  AGD.  If  the  lines  do  separate  at  (7,  one 
or  the  other  must  change  direction  at  this  point ;  but  this 
is  contradictory  to  the  definition  of  a  straight  line  (D.  4) : 
hence,  the  supposition  that  they  separate  at  any  point  is 
absurd.  They  must,  therefoi'e,  coincide  throughout;  which 
was  to  be  proved. 

Cor.     Two  straight  lines  can  intersect  in  only  one  point. 

Note. — The  method  of  demonstration  employed  above, 
called  the  reductio  ad  ahsurdum.  It  consists  in  assuming  an 
hypothesis  which  is  the  contradictory  of  the  proposition  to 
be  proved,  and  then  continuing  the  reasoning  lantil  tlie 
assumed  hypothesis  is  shoAvn  to  be  false.  Its  contradictory  is 
thus  proved  to  be  true.  This  method  of  demonstration  id 
ofi.en  used   in   Geometry. 


94  GEOMETRY. 


PROPOSITION     IV.      THEOEEM. 

If  a  straight  line  meet  two  other  straight  lines  at  a  com 
mon  ^x>2wZ,  mahing  the  sum  of  the  contiguous  angles 
equal  to  two  right  angles,  the  two  lines  met  will  foim 
one  and  the  same  straight  line. 

Let  DC  meet  AG  and  BG 
at  C,  making  the  sum  of  the 
angles  DGA  and  DCB  equal 
to  two  right  angles :  then  avlII 
GB    be   the  jn-olongation   of  AG. 

For,  if  not,  suppose  GE  to  be  the  prolongation  of  AC; 
then  will  the  sum  of  the  angles  DGA  and  DGE  be 
equal  to  two  right  angles  (P.  I.)  :  We  shall,  consequently, 
have     (A.   1), 

DGA  +  DGB  =  DGA  +  DGE  ; 

Taking   from    both    the    common    angle    DGA,     there    re- 


mains. 


BGB  =  DGE, 


which  is  impossible,  since  a  part  cannot  be  equal  to  the 
whole  (A.  8 ).  Hence,  GB  must  be  the  prolongation  of 
AG  ;    which   was  to  be  lyroved. 


PROPOSITION     V.       THEOREM. 

If   two   triangles  have    tioo    sides   and  the  included  angle  of 

the    one    equal    to    two    sides    and    the  included  angle  of 

the  other,  each  to  each,  the   triangles  loill  he  equal  in  aU 
their  parts. 

In    the    triangles    ABG    and     DEF,      let    AB    be    equal 


BOOK    I. 


25 


to  DE^  AG  to  JDF^  and  the  angle  A  to  the  angle  B : 
then   will  the   triangles   be   equal   in  all   their  parts. 

For,    let    ABC    be 
applied    to    BEF^     in  A  D 

such  a  manner  that  the 
angle  A  shall  coincide 
with  the  angle  Z>, 
the  side  AE  taking 
the   direction  DE^    and 

the  side  AC  the  direction  BF.  Then,  because  AB  is 
equal  to  BE^  the  vertex  B  will  coincide  with  the  vertex 
E\  and  because  AC  is  equal  to  BF^  the  vertex  G  will 
coincide  with  the  vertex  F\  consequently,  the  side  BG 
will  coincide  with  the  side  EF  (A.  11).  The  two  triangles, 
therefore,  coincide  throu'diput,  and  are  consequently  equal  in 
all  their  parts     (I.,   D.  44)  ;    which  was  to   be  proved. 


PROPOSITION"     VI.      TIIEOPvEM. 

J^  two  triangles  have  two  angles  and  the  included  side  of  the 
07ie  equal  to  two  angles  and  the  included  side  of  the  other, 
each  to  each,  the  triangles  xcill  be  equal  in  all  their  parts. 

In  the  triangles 
ABC  and  BEE,  let 
the  angle  B  be  equal 
to  the  angle  E,  the 
angle  C  to  the  angle 
F,  and  the  side  BG 
to  the  side  EF:  then 
will    the   ti'iangles   be   equal  in   all   their   parts. 

For,    let    ABC    be    applied    to   BEE    in   such   a   manner 
that  the   angle   B    shall  coincide  with   the  angle   E,     the  side 


26  GEOMETRY. 

BiJ  taking  the  direction  EF,  and  the  side  JBA  the  direc- 
tion ED.  Then,  because  BG  is  equal  to  EF^  the  vertex 
C  will  coincide  with  the  vertex  F\  and  because  the  angle 
C  is  equal  to  the  angle  F^  the  side  CA  will  take  the 
direction  FD.  Now,  the  vertex  A  being  at  the  same  tinio 
on  the  lines  ED  and  FD^  it  must  be  at  their  intersection 
D  (P.  in.,  C.)  :  hence,  the  triangles  coincide  throug^ojat, 
and  are  therefore  equal  in  all  their  parts  ( I.,  D.  -^(4^)  ; 
which  was  to  he  proved. 


PROPOSITION    Vn.      THEOREM. 

The  sum  of  any  two  sides  of  a   triangle  is   greater  than  the 

third   side. 

Let  AB  C  be  a  triangle  :  then  will 
the  sum  of  any  two  sides,  as  AB,  BC, 
be   greater   than   the   third   side    AC. 

For,  the  distance  from  A  to  C, 
measured   on   any  broken   line    AB,  BC, 

is  greater  than  the  distance  measured  on  the  straight  line 
AC  (A.  12)  :  hence,  the  sum  of  AB  and  i>C  is  greater 
than    AC  ;    which  was   to   be  proved. 

Cor.     K  from   both   members   of   the   inequality, 

AC  <AB  +  BC, 

we  take  away  either  of  the  sides  AB,  BC,  as  BC,  for 
example,   there   will   remain    (A.  5), 

AC  -  BC<AB; 

that  is,  the  difference  hetween  any  two  sides  of  a  triangle  is 
less  than  the  third  side. 

Scholium.    In    order    that    any  three    given    lines  may   re- 


BOOK    I. 


27 


present  the  sides  of  a  triangle,  tlie  sum  of  any  two  must  be 
greater  tban  the  third,  and  the  difference  of  any  t^vo  must 
be  less  than  the   thii'd. 


PROPOSITION     Vni.       THEOEEM. 

If  from    any  'point  within   a   triangle    t^oo    straight  lines   h 
drawn  to   the    extremities    of    any    side,   their  sum  icill  be 
less  than,  that  of  the  two  remaining  sides  of  the  t7'iangle. 

Let  0  be  any  point  within  the  triangle  JBAC,  and  let 
the  lines  OJ^,  00,  be  drawn  to  the 
extremities  of  any  side,  as  BC  '. 
then  will  the  sum  of  DO  and  OC 
be  less  than  the  sum  of  the  sides 
BA    and    AC. 

Prolong   one    of    the   lines,    as  BO, 
till  it  meets  the  side   AC    in    D;    then,  from  Prop.  VII.,  we 
shall   have, 

OC  <  (9Z>  +  DC  ; 

adding    BO    to   both   members    of  this   inequality,  recollecting 

that  the  sum  of  BO  and  OD  is  equal  to  BB,  we  have 
(A.  4), 

BO  +0C  <  BD  +  BC. 

From   the   triangle    BAB,     we   have    (P.  VII.), 

BB  <  BA  +  AB  ; 

adding  BC  to  both  members  of  this  inequality,  recollecting 
that  the   sum   of   AD    and   DC    is   equal   to   AC,    we   have, 

BD  +  DC<  BA  +  AC. 

But  it  was  shown  that  BO -\-  OC  is  less  than  BD-\DC; 
still  more,  then,  is  BO  +  OC  less  tlan  BA  +  AC  ;  lehich 
was   to  be  jyroved. 


\ 


28 


GEOMETRY. 


PEOPOSmOX     IX.       THEOREM. 


If  two  triangles  have  two  sides  of  the  on^  equal  to  two  sides  of 
the  other,  each  to  each,  and  the  included  angles  unequal,  the 
third  sides  will  be  unequal ;  and  the  greater  side  will  belong 
to  the  triangle  which  /uis  the  greater  included  angle. 

In  the  triangles  BA  C  and  DEF,  let  AB  be  equal  to 
DE,  AG  to  BE,  and  the  angle  A  greater  than  the  an- 
gle   B :     then  will    BG    be  greater  than    EE. 

Let  the  line  AG  be  dra-wn,  makirisc  the  ansrle  GAG 
eqnal  to  the  angle  B  (Post.  7)  ;  make  AG  equal  to  BE, 
and  draw  GG.  Then  will  the  triangles  AGG  and  BEE 
have  two  sides  and  the  included  angle  of  the  one  equal  to 
two  sides  and  the  included  angle  of  the  other,  each  to  each; 
consequently,     GG    is   equal    to    EF    (P.  V.). 

Xow,  the  point  G  may  be  without  the  triangle  ABG^ 
it  may  be  on  the  side  BG,  or  it  may  be  within  the  tri- 
angle   ABG.      Each  case  will  be  considered  separately. 

A  D 

1°.       When     G     is 

without     the     triangle 

ABG. 

In  the  triangles  GIG 

and    AIB,     we    have, 

(P.  \^I.), 

GI+  IG  >  GG,      and      BI  +  lA  >  AB  ; 

whence,  by  addition,  recollecting  that  the  sura  of  BI  and 
IG  is  equal  to  BG,  and  the  sum  of  GI  and  lA,  to  GA, 
we  have, 

AG  +  BG  >  AB  ^  GG. 


BOOK    I. 


29 


Or,   since    AG  =  AB,    and     GC  =  EF,    we  have, 
AB  +  BC  >  AB  +  EF, 
Taking   away  the    common   part  AB^     there    remams     (A-  5), 

BC  >  EF. 

2\     Wlien    G    is  on    BC. 

In  this  case,  it  is  obvious 
that  GC  is  less  than  BC\  or, 
since    GC  =  EF^     we  have, 

BC  >  EF. 

3°.     When    G    is  witliin   the  triangle    ABC. 
From   Proposition  Vlil.,  we  have, 

BA  +  BC  >  GA  +  GG; 

or,  since     GA  =  BA,     and    GC  =  EF, 

wo  have, 

BA  +  BC  >  BA  +  EF. 

Taking     away    the    common    part     AB, 

there  remains, 

BC  >  EF 

Hence,  in  each  case,  BC  is  greater  than  EF'y  which  teas 
to  he  proved. 

Conversely:  If  in  two  triangles  ABC  and  BEF,  the 
BJde  AB  is  equal  to  the  side  BE,  the  side  AC  to  BE, 
and  B  C  greater  than  EF,  then  wUl  the  angle  BA  C  bo 
scaler  than  the  angle    EDF. 

For,  if  not,  BAC  must  either  be  equal  to,  or  less  than, 
EDF.  In  the  former  case,  BC  would  be  equal  to  EF 
(P.  v.),  and  in  the  latter  case,  BC  would  be  less  tlian 
EF\  either  of  which  would  be  contrary  to  tlie  hj-pothesis  : 
hence,    BAC    must  be  greater  than    EDF. 


30 


GEOMETRY. 


PEOPOSITION     X.       THEOllEM. 

If  tico  triangles  have  the  three  sides  of  the  one  equal  to  the 
three  sides  of  the  other,  each  to  each,  the  triangles  loill  b€ 
equal  in  aU  their  parts. 

In  tlie  triangles  ABC  and  DEF,  let  AB  be  equal  to 
DE,  AC  to  BF,  and  BC  to  FF :  then  will  the  tri- 
angles  be  equal  in  all  their  parts. 

For,   since   the    sides 
AB,  AC,    are  equal  to  A  D 

BF,  BF,  each  to  each, 
if  the  angle  A  were 
greater  than  B,  it  would 
follow,  by  the  last  Pro- 
position,   that    the    side 

BC  would  be  greater  than  FF;  and  if  the  angle  A  were 
less  than  B,  the  side  BC  would  be  less  than  FF.  But 
BC  is  equal  to  FF,  by  hypothesis  ;  therefore,  the  angle  A 
can  neither  be  greater  nor  less  tlian  B  :  hence,  it  must  be 
equal  to  it.  The  tA\^o  triangles  have,  therefore,  two  sides  and 
the  included  angle  of  the  one  equal  to  two  sides  and  the  inclu- 
ded angle  of  the  other,  each  to  each  ;  and,  consequently,  they 
are   equal  in  all  their  parts   (P.  V.)  ;   which  was  to  be  proved. 

Scholium.     In   triangles,   equal   iji   all   their  parts,    the    equal 
sides  lie  opposite  the  equal  angles;   and   conversely. 


rEOPOSITION     XI.      THEOEEM. 

In  an  isosceles  trimigle  the  angles  opposite  the  equal  sides  are 

equal. 

Let  BAC  be  an  isosceles  triangle,  having  the  side  AB 
equal  to  the  side  AC',  then  will  the  angle  C  be  equal  to 
the  angle    B. 


BOOK    I. 


31 


Join  the  vertex  A  and  the  middle  point  D  of  the  base 
JiC  Then,  AB  is  equal  to  AC,  by  hypothesis,  AD 
common,  and  JjJ)  equal  to  DC,  by 
construction  :  hence,  the  triangles  DAD, 
and  DAC,  have  the  three  sides  of  the 
one  equal  to  those  of  the  other,  each  to 
each  ;  therefore,  by  the  last  Proposition, 
the  angle  D  is  equal  to  the  angle  C ; 
ichich  was  to  be  "proved. 

Cor,    1.     An   equilateral  triangle  is   equiangular. 

Cor.  2.  The  angle  DAD  is  equal  to  DAC,  and  DDA 
to  CDA  :  hence,  the  last  two  are  right  angles.  Conse- 
quently, a  straight  line  draioji  from  the  vertex  of  an  isosceles 
triangle  to  the  middle  of  the  base,  bisects  the  angle  at  the  vertex, 
and  is  perpendicular  to  the  base. 


PEOPOSITIOl^J"     XII.      THEOEEM. 

^  two  angles  of  a  triangle  are  eqiial,  the  sides  opposite  to 
them  are  also  equal,  and  conseqiiently,  the  triangle  is  isos- 
celes. 

In  the  triangle  ADC,  let  the  angle 
ADC  be  equal  to  the  angle  ACD : 
then  will  AC  be  equal  to  AD,  and 
consequently,  the  triangle  will  be  isosceles. 

For,  if  AD  and  AC  are  not  equal, 
suppose  one  of  them,  as  AD,  to  be  the 
greater.  On  this,  take  DD  equal  to  AC  (Post.  3),  and 
draw  DC.  Then,  in  the  triangles  ADC,  DDC,  we  have 
the  side  DD  equal  to  AC,  by  construction,  the  side  DO 
common,  and  the  included  angle  ACD  equal  to  the  included 
angle  DDC,  by  hypothesis:    hence,  the  two  triangles  are  equal 


/■ 


32 


GEOMETRY. 


in  all  their  parts  (P.  V.).  But  this  is  impossible,  because  a 
part  cannot  be  equal  to  the  whole  (A.  8)  :  hence,  the 
hypothesis  that  AJ3  and  AC  are  unequal,  is  false.  They 
must,   therefore,   be   equal ;    which  was   to  be  proved. 

Cor.     An   equiangular  triangle  is   equilateral. 


PROPOSITION     Xin.       THEOREM. 

In  any  triangle^  the  greater  side  is  op2^osite  the  greater  angle; 
and^  conversely^  the  greater  angle  is  opposite  the  greater 
$ide. 

In  the  triangle  ABC^  let  the  angle 
ACB  be  greater  than  the  angle  ABC: 
then  will  the  side  AB  be  greater  than 
the   side    A  G. 

For,  draw  (7i>,  making  the  angle 
BCD    equal  to  the  angle   B     (Post.  7) : 

then,  in  the  triangle  BGB,  we  have  the  angles  BCB  and 
BBC  equal:  hence,  the  opposite  sides  J)B  and  DC  are 
equal   (P.  XII.).      In  the  triangle    ACB,    we  have  (P.  VIE.), 

AD  +  DC  >  AC  ', 
or,  since    DC  =  DB,    and    AD  +  DB  =  AB,    we  have, 

AB>  AC  \ 

which  teas  to  be  proved. 

Conversely  :  Let  AB  be  greater  than  A  C :  then  Aviil  t],e 
angle  ACB    be   greater  than  the   angle    ABC. 

For,  if  ACB  were  less  than  ABC,  the  side  AB  would 
be  less  than  the  side  A  C,  from  what  has  just  been  proved  ; 
if  ACB  were  equal  to  ABC,  the  side  AB  would  be 
equal  to  AC,  by  Prop.  XII.;   but  both  conclusions  are  contrary 


BOOK    I. 


33 


to  the  hypothesis :  hence,  A  CB  can  neither  be  less  than, 
nor  equal  to,  AB  G  ;  it  must,  therefore,  be  greater ;  which 
wae   to  be  proved. 


B     E 


PROPOSITION     XrV.      THEOPvEM.      . 

I'Votn   a  given  point  onJij  one  perpendicular  can  he  drawn  U 
a  given  straight  li?ie. 

Let  ^  be  a  given  point,  and  AB 
a  perpendicular  to  J)£J :  then  can  no 
other  perpendicular  to  DJEJ  be  drawn 
from  A. 

For,  suppose  a  second  perpendicular 
AC  to  be  drawn.  Prolong  AB  till 
BJF  is  equal  to  AB,  and  draw  CF. 
Then,  the  triangles  ABC  and  FBC  will  have  AB  equal 
to  BF,  by  constructioii,  CB  common,  and  the  included 
angles  ABC  and  FBG  equal,  because  both  are  right  an- 
gles  :  hence,  the  angles  A  CB  and  JP'CB  are  equal  (P.  V.) 
But  ACB  is,  by  a  hjqDOthesis,  a  right  angle  :  hence, 
Ji'CB  must  also  be  a  right  angle,  and  consequently,  the  line 
ACJP  must  be  a  straight  line  (P.  IV.).  But  this  is  impos- 
sible (A.  11).  The  hypothesis  that  two  perpendiculars  can 
be  drawn  is,  therefore,  absurd ;  consequently,  only  one  such 
perpendicular   can   be   drawn  ;    which   was   to   be  proved. 

If  the  given  point  is  on  the  given  line,  the  proposition 
is  equally  true.  For,  if  from  A  two  perpendiculars  AB 
and  A  C  could  be  drawn  to  BU, 
we  should  have  BAJS  and  CA^ 
each  equal  to  a  right  angle  ;  and 
consequently,  equal  to  each  other  ; 
which   is   absurd    (A.  8). 


34 


tlEOMETRY. 


PROPOSITION     XV.      THEOREM. 

If  from  a  point  without  a  straight  li?ie  a  perpendicular  le 
let  fall  on  the  line,  and  oblique  lines  be  drawn  to  differ- 
ent points  of  it : 

1°.     The  perpendicular  xcill  be  shorter  than  any  oblique  line. 

2°.  Ally  tico  oblique  lines  that  meet  the  given  line  at  points 
equally  distant  from  the  foot  of  the  perpendicular,  wiU 
be  equal: 

3°.  Of  tioo  oblique  lines  that  meet  the  given  line  at  points 
unequally  distant  frojn  the  foot  of  the  perpendicular,  the  one 
which  meets  it   at   the  greater  distance  will  be  the  longer. 

Let  J.  be  a  given  point,  DE  a 
given  straight  line,  AB  a  perpendicular 
to  DE,  and  AD,  AC,  AE  oblique 
lines,  BG  being  equal  to  BE,  and  BD 
greater  than  BC.  Then  will  AB  be 
less  than  any  of  the  oblique  lines,  AC 
will  be  equal  to  AE,  and  AD  greater 
than    A  C. 

Prolong    AB     untU     BE     is    equal    to 
FC,    ED. 

1°.  In  the  triangles  ABC,  EBC,  we  have  the  side 
AB  equal  to  BE,  by  construction,  the  side  BC  common, 
and  the  included  angles  ABC  and  EBC  equal,  because  both 
are  right  angles:  hence,  EC  is  equal  to  AC  (P.  V.)- 
But,  AE  is  shorter  than  ACE  (A.  12):  hence,  AB,  the 
half  of  AE,  is  shorter  than  AC^  the  half  of  ACE',  which 
was   to   be  proved. 

2°.  In  the  triangles  ABC  and  ABE,  we  have  the 
side  BC  equal  to  BE,  by  hypothesis,  the  side  AB  com 
men,    and    the     included    angles    ABC     and     ABE     equal, 


AB,     and    draw 


BOOK    I. 


35 


because  both  arc  right  angles:   "hence,   AC    is  c:<iii;il  to   A^; 
which  teas   to   be  proved. 

3°.  It  may  be  sho'mi,  as  in  the  first  case,  that  AD  is 
equal  to  DK  Then,  because  the  point  C  lies  witliin  the 
triangle  ADJ^,  the  sum  of  the  lines  AD  and  DF  will  be 
greater  than  the  sum  of  the  lines  A  C  and  CF  (P.  Vm.) : 
hence,  AD,  the  half  of  ADI]  is  greater  than  AC^  the 
half  of   A  CF ;    which  loas   to  he  proved. 

Cor.  1.  The  perpendicular  is  the  shortest  distance  from  a 
point   to   a   line. 

Cor.  2.  From  a  given  point  to  a  given  straight  line,  only 
two  equal  straight  lines  can  be  drawn  ;  for,  if  there  could 
be  more,  there  would  b^  at  least  two  equal  oblique  lines  on 
the  same  side  of  the  perpendicular  ;   which  is  impossible. 


PROPOSITION    xYi.     theorem:. 

If  a  perpendicidar  he  drmon   to   a  given  straight  line   at  its 

middle  point : 
1°.     A7ig  point  of   the  perpendicular  wiU  he    equally   distant 

from   the  extremities  of  the  line: 
2°.     Any  point,   without   the  perpendicidar,   will  he    unequally 

distant  from  the  extremities. 

Let  AB  be  a  given  straight  line,  G 
its  middle  point,  and  FF  the  perpendicular. 
Then  will  any  point  of  EF  be  equally  dis- 
tant from  A  and  B  ;  and  any  point  without 
EF,  will  be  unequally  distant  from  A  and  B. 

1°.     From  any  point  of  EF,   as  D,  draw 
the  lines    DA     and    DB.       Then   Anil   DA 
and   DB  be  equal  (P.  XV.)  :    hence,    D   is 
equally   distant  from    A     and    B  ',    which  was   to   he  proved. 


36 


GEOMETRY. 


2®.  From  any  point  -witliout  EF^  as  Z,  draw  lA  and 
IB.  One  of  these  lines,  as  il-J,  will  cut  EF  in  Bomo 
point  I) ;  draw  BB.  Then,  from  what 
has  just  been  shown,  BA  and  DB  will 
be  equal  ;  but  IB  is  less  than  the  sum 
of  ID  and  DB  (P.  VII.)  ;  and  because 
the  sum  of  ID  and  DB  is  equal  to  the 
sum  of  ID  and  DA^  or  Z4,  we  have 
/B  less  than  lA  :  hence,  I  is  unequally 
distant  from  A  and  i?  ;  which  was  to  he 
proved. 

Cor.  If  a  straight  line  EF  have  two  of  its  points  E 
and  F  equally  distant  from  A  and  B,  it  will  be  perpen- 
dicular to   the   Une    AB    at  its   middle   point. 


PKOPOSITION     XVn.       THEOREM. 

If  two  right-angled  triangles  have  the  hypotJiennse  and  a  side 
of    the   one  equal  to  the   hypothemise   and   a  side  of  the 
other^  each  to  each^  the  triangles  will  be  equal  in  all  their 
parts. 
Let    the    right-angled    tri- 

andes  AB  C  and  DEF  have 

the    hypothenuse    AC    equal 

to    DF,    and  the    side    AB 

equal  to    DE:   then  will  the 

triangles  be  equal  in  all  their  parts. 
K  the   side    BC    is  equal   to     EF, 

equal,  in  accordance  with  Proposition  X. 

that    BG     and     EF     are     imequal,     and    that     BC     is    the 

longer.       On    BG     lay   off    BG      equal   to   EF,      and    draw 

AG.      The  triangles    ABG    and    DEF   have    AB    equal  to 

DE,  by  hypothesis,   BG    equal  to    EF,    by  construction,  and 


the   triangles  will    be 
Let  us  suppose  then, 


BOOK    I. 


37 


the  angles  JB  and  E  equal,  because  both  are  right  angles ; 
consequently,  AG  is  equal  to  DF  (P.  V.)  But,  AC  is 
equal  to  DJF^  by  hypothesis:  hence,  AG  and  AC  are  equal, 
which  is  impossible  (P.  XV.).  The  hypothesis  that  BC  and 
EF  are  unequal,  is,  therefore,  absurd  :  hence,  the  triangles 
have  all  their  sides  equal,  each  to  each,  and  are,  consequently, 
equal  in  all  of  their  parts  ;   which  was   to  be  proved. 

PROPOSITION    XVni.       THEOREM. 

Jf  two  straight  lines  are  perpendicular  to  a  third  straight  linSf 

tliey  loill  he  parallel. 

Let  the  two  lines     A  C,    -SZ>,     be  perpendicular  to  AB : 
then  will  they  be  parallel. 

For,  if  they  could  meet  in  a 
point  0,  there  would  be  two 
perpendiculars  OA^  OB,  drawn 
from  the  same  point   to  the  same 

straight  Une  ;  which  is  impossible  (P.  XIV.)  :  hence,  the 
lines  are  parallel ;    which  was   to  be  p>rov€d. 


D 


-O 


DEFESfinONS. 

If  a  straight  line  EF  inter- 
sect two  other  straight  lines  AB 
and  CD,  it  is  called  a  secant, 
with  respect  to  them.'  The  eight 
angles  formed  about  the  points  of 
intersection  have  diflerent  names, 
with   respect   to   each   other. 

1°.  Interior  angles  on  the  same  side,  are  those  that 
lie  on  the  same  side  of  the  secant  and  within  the  other  two 
lines.  Thus,  BGH  and  GHD  are  interior  angles  on  the 
same  side. 


38  GEOMETRY. 

2°.     Exterior  angles  on  the  same  side,  are  those  that  lie 
on  the   same   side   of  the   secant   and  without    the    other    two 
lines.      Thus,    EGB    and  DHF 
are   exterior  angles  on    the   same 
side. 

3°.  Alternate  angles,  are 
hose  that  lie  on  opposite  sides 
t)f  the  secant  and  loithhi  the 
other  two  lines,  but  not  adja- 
cent. Thus,  AGH  and  GRD 
are  alternate  angles. 

4°.  Alternate  exterior  angles,  are  those  that  lie  on 
opposite  sides  of  the  secant  and  without  the  other  two  lines. 
Thus,    AGE    and    FHD    are  alternate  exterior   angles. 

5°.  Opposite  exterior  and  interior  angles,  are  those 
that  lie  on  the  same  side  of  the  secant,  the  one  within  and 
the  other  without  the  other  tAVO  lines,  but  not  adjacent.  Thus, 
EGB    and    GHD    are    ojiposite   exterior   and  interior  angles. 


PEOPOSITION     XIX.       THEOREM, 

If  two  straight  lines  meet  a  third  straight  line,  making  the 
sum  of  the  interior  angles  on  the  same  side  equal  to  two 
right  angles,   the  tivo   lines  will  be  parallel. 

Let  the  lines  EC  and  ED  meet  the  line  BA,  maldng 
the  sum  of  the  angles  BAC  and  ABB  equal  to  two  right 
angles :    then  will    EC    and    EB    be  parallel. 

Through  G,    the  middle  point 
of  AB^    draw  GE  perpendicular  -g  g 

to    EC,     and  prolong  it    to    E. 


The  sum  of  the  anp:les   GBE 


G 


and   GBB    is  equal  to  two  right       ^  aF  ^ 


i 


BOOK    I.  Zy 

angles   (P.  I.)  ;    tlie   sum   of  the  angles   FAG     and    GBD    is 
equal   to   two   right  angles,  by  hypothesis  :    hence    (A.  1), 

GJiU+  GBB  r=  FAG  +  GBD. 

Taking  from  both  the  common  part  GBD,  we  have  the 
angle  GBE  equal  to  the  angle  FAG.  Again,  the  angles 
BGF  and  AGF  arc  equal,  because  they  are  vertical  an- 
gles (P.  II.)  :  hence,  the  triangles  GFB  and  GFA  have 
two  of  their  angles  and  the  hicludcd  side  equal,  each  to  each  ; 
they  are,  thci-efore,  equal  in  all  their  parts  (P.  VI.) :  hence, 
the  angle  GEB  is  equal  to  the  angle  GFA.  But,  GFA 
is  a  right  angle,  by  construction  ;  GEB  must,  therefore,  be 
a  rio-ht  anirle  :  hence,  the  Imes  KG  and  SD  are  both  per- 
pendicular  to  EF^  and  are,  therefore,  parallel  (P.  XVm,)  ; 
which  xoas   to   he  2^oved. 

Cor.  1.  If  two  straight  lines  arc  cut  by  a  third  straight 
line,  making  the  alternate  angles  equal  to  each  other,  the 
two   straight  lines   will   be   parallel. 

Let  the  angle  HGA  be  equal 
to  GUD.  Adding  to  both,  the 
angle    IIGB,     we  have, 

HGA  +  HGB  =  GIID  +  IIGB. 

But    the    first    sum    is    equal    to 

two   right   angles   (P.  I.)  :    hence, 

the   second  sum   is   also   equal  to   two  right  angles  ;    therefore, 

from  what  has  just  been  sIiowtd,    AB    and     CD    are  parallel. 

Cor.  2.  If  two  straight  lines  are  cut  by  a  third,  making 
the  opposite  exterior  and  interior  angles  equal,  the  two  straight 
lines  will  be  parallel.  Let  the  angles  EGB  and  GHD  be 
equal :  Now  EGB  and  A  GH  are  equal,  because  they  are  verti- 
cal angles  (P.  II.);  and  consequently,  AGU  and  GHD  are 
equal:    hence,  from   Cor.  1,  AB  and   CD  are  parallel. 


40  GEOMETRY. 


PKOPOBITIOIT     XX.       THEOREM. 

If  a  straight  line  intersect  two  parallel  straight  lines,  the  sum 
of  the  Ulterior  angles  on  the  same  side  will  he  equal  to 
tioo  right   angles 

Let  the  parallels  AB,  CD,  be  cut  by  the  secant  jine 
FE :  then  will  the  sum  of  IIGB  and  GITD  be  equal  to 
two   rioht   angles. 

For,  if  the  sum  of  IIGB 
and  GJID  is  not  equal  to 
two  right  angles,  let  JGJO  be 
drawn,  making  the  sum  of  IIGL 
and  GIID  equal  to  two  right 
angles  ;  then  IL  and  CD  will 
be  parallel  (P.  XIX.)  ;  and  consequently,  we  shall  have  two 
lines  GB,  GL,  draAvn  through  the  same  point  G  and  par- 
allel to  CD,  which  is  impossible  (A.  13)  :  hence,  the  sum 
of  HGB  and  GIID,  is  equal  to  two  right  angles  ;  which 
was   to   he  proved. 

In  like  manner,  it   may  be  proved   that  the   sum  of   HGA 
and    GHC,    is   equal   to   two   right  angles. 

Cor.  1.  If  HGB  is  a  right  angle,  GIID  will  be  a  right 
angle  also  :  hence,  if  a  line  is  perpendicular  to  one  of  two 
parallels,  it  is  perpendicular  to   the  other  also. 

Cor.  2.  If  a  straight  line  meet  ticc  parallels,  the  alternate 
angles  will  he  equal 

For,  if  AB  and  CD  are 
parallel,  the  sum  of  BGII  and 
GIID  is  equal  to  two  right 
angles  ;  the  sum  of  BGII  and 
HGA  is  also  equal  to  two  right 
angles  (P.  I.)  :  hence,  these  sums 


BOOK    I.  41 

are  eqaal.  Taking  away  the  common  part  BGH^  there  re- 
mains the  angle  GIID  equal  to  JIGA.  In  like  manner, 
it  may  be  shown  that    BGH    and    GUC    are  equal. 

Cor.  Z.  If  a  straight  line  meet  two  parallels^  t/ie  ojijiositt 
exterior  and  interior  angles  will  he  equal.  The  angles  DUG 
and  HGA  are  equal,  from  Avhat  has  just  been  shown.  The 
angles  JIGA  and  BGE  are  equal,  because  they  are  verti- 
cal :  hence,  DUG  and  BGE  are  equal.  In  like  manner, 
it  may  be   shown   that     CIIG    and    AGE    are   equal. 

Scholium.  Of  the  eight  angles  formed  by  a  line  cutting 
two  parallel  lines  obliquely,  the  four  acute  angles  are  equal, 
and   so,   also,    are   the  four  obtuse   angles. 


pkopositio:n"    xxi.     theorem. 

Jf  two  straight  lines  intersect  a  third  straight  line,  making  the 
sum  of  tlie  interior  angles  on  the  same  side  less  than  ttvo 
right  anr/les,  the  two  lines  loill  meet  if  sufficiently  produced. 

Let  the  two  lines  CZ>,  7Z,  meet  the  line  EF^  making 
the  sum  of  the  interior  angles  UGL^  GHD^  less  than  two 
right  augles  :  then  will  IL  and  CD  meet  if  suflSciently  pro- 
duced. 

For,  if  they  do  not  meet, 
they  must  be  parallel  (D.  16). 
But,  if  they  were  parallel,  the 
sum  of  the  interior  angles  HGL^ 
GUD,  would  be  equal  to  two 
right  angles  (P.  XX.),  which  is 
contrary  to  the  hypothesis  :  hence, 
iZ,  CD^  M'ill  meet  if  sufficiently  produced  ;  which  was  to  be 
proved. 


42 


GEOMETRY. 


Cor.  It  is  evident  tliat  IL  and  CB  will  meet  on  tliat 
side  of  EF,  on  which  the  sum  of  the  two  angles  is  less 
than   two   right  angles. 


PROPOSITION     XXII.      THEOREM. 

Tf  two    Straight  lines  are  parallel  to   a   third  line,   they  arc. 

parallel  to  each  other. 

Let  AB  and  CB  be  respectively- 
parallel  to  EF:  then  will  they  be  par- 
allel  to   each   other. 

For,  draw  PB  perpendicular  to 
EF\  then  will  it  be  perpendicular  to 
AB,  and  also  to  CB  (P.  XX.,  C.  1) : 
hence,  AB  and  CB  are  perpendicu- 
lar to  the  same  straight  hne,  and  consequently,  they  are  par 
allel  to   each   other    (P.  XVIII.)  ;    which  was   to   he  proved. 


E 

R 

F 

C 

Q 

D 

A 

p 

B 

PROPOSITION     XXIII.       THEOREM. 
Two  parallels   ate  everywhere  equally   distant. 

Let    AB    and    CB    be  parallel :    then   will  they  be  every- 
where  equally  distant. 

From  any  two  points  of  AB,  as 
F  and  E,  draw  FH  and  EG 
perpendicular  to  CB  ;  they  will  also  be 
perpendicular  to  AB  (P.  XX.,  C.  1), 
and  will  measure   the  distance  between 

AB  and  CB,  at  the  points  F  and  E.  Draw  also  VG 
The  hnes  FIT  and  EG  are  parallel  (P.  XTIII.)  :  hence, 
the  alternate  angles  E:FG  and  FGE  are  equal  (P.  XX.,  C.  2). 
The   lines  AB    and    CB    are   parallel,   by  hypothesis :    hence, 


BOOK    I.  48 

the  alternate  angles  EFG  and  FGH  are  equal.  The  tii- 
angles  FGE  and  FGH  have,  therefore,  the  angle  JIGF 
equal  to  GFF,  GFII  equal  to  FGF.  and  the  side  FG 
common  ;  they  are,  therefore,  equal  in  all  their  parts  \V.  VI.) 
hence,  FH  is  equal  to  FG  ;  and  consequently,  AB  and 
CD    are  everywhere  equally  distant ;  wliich  was  to  he  proved^ 


PROPOSITION     XXIV.      THEOEEM. 

If  two  angles  have  their  sides  parallel^   and  lying  either  in 
the  same,  or  in  opposite  directions^  they  will  he  equal. 

1°.  Let  the  angles  ABC  and  DEF  have  their  sides 
parallel,  and  lying  in  the  same  du-ection  :  then  will  they  be 
equal. 

Prolong    FF    to    L.      Then,   because  ^/  ij^ 

BE    :aid    AL    are  parallel,   the   exterior  /  / 

angle    DEF    is  equal  to   its    opposite  in-        L/- -d^ F 

terior   angle   ALE    (P.  XX.,  C.  3)  ;   and        / 
because    BC     and    LF    are  parallel,  the 
exterior   angle   ALE    is   equal  to   its  op- 
posite    interior    angle    ABC  :      hence,     BEF    is    equal    to 
ABC  ;    which  was   to   he  p>ro'oed. 

2°.  Let  the  angles  ABC  and  GHK 
have  their  sides  parallel,  and  lying  in  op- 
posite directiora  :   then  Mill  they  be  equah 

Prolong  GH  to  M.  Then,  because 
KH  and  BJSl  are  parallel,  the  exterior 
angle  GIIIZ  is  equal  to  its  opposite  interior  angle  HMB , 
and  because  HM  and  BC  are  parallel,  the  angle  HMB 
is  equal  to  \vi  alternate  angle  MBO  (I'.  XX.,  C.  2)  :  hence, 
QHK   is   equitl  to    AB  C ;     which  icas   to  he  proved. 

Cor.    The   C]'.\K.«to  angles   of  a  parallelogram  are   equal 


/'  ii 


M  GEOMETRY. 


PROPOSITION    XXV.      THEOREM. 

In  any  triangle^  the  sum  of  the  three  angles  is  equal  to   two 

right  angles. 

Let     CBA     be    any  triangle :     then   Avill    the   sum    of   tlie 
angles    (7,     A^     and    B^    be  equal  to 
two  right   angles. 

For,  prolong  CA  to  Z>,  and  draw 
AE    parallel  to    BC. 

Then,  since  AE  and  CB  are 
parallel,  and  CD  cuts  them,  the  ex 
terior    angle    DAE     is    equal    to    its 

opposite  interior  angle  C  (P.  XX.,  C.  3).  In  like  manner, 
since  AE  and  CB  are  parallel,  and  AB  cuts  them,  the 
alternate  angles  AB  C  and  BAE  are  equal :  hence,  the 
sum  of  the  three  angles  of  the  triangle  BAC^  is  equal  to 
the  sura  of  the  angles  CAB,  BAE,  EAB  ;  but  this  sum 
is  equal  to  two  right  angles  (P.  I.,  C.  2);  consequently,  the 
sum  of  the  three  angles  of  the  triangle,  is  equal  to  two 
right   angles  (A.  1)  ;    which  was   to   be  proved. 

Cor.  1.  Two  angles  of  a  triangle  being  given,  the  thii-d 
will  be  found  by  subtracting  their  sum  from  two  right  angles. 

Cor.  2.  If  two  angles  of  one  triangle  are  respectively 
equal  to  two  angles  of  another,  the  two  triangles  are  mutually 
equiangular. 

Cor.  3.  In  any  triangle,  there  can  be  but  one  right  angle  j 
for  if  there  were  two,  the  third  angle  would  be  zero.  Nor 
can  a  triangle  have  more  than  one  obtuse  angle. 

Cor.  4.  In  any  right-angled  triangle,  the  sum  of  the  acute 
angles  is  equal  to  a  right  angle. 


BOOK    I.  45 

Cor.  5.  Since  every  equilateral  triangle  is  also  equianguiar 
(P.  XI.,  C.  1),  each  of  its  angles  will  be  equal  to  the  third  part 
of  two  right  angles  ;  so  that,  if  the  right  angle  is  expressed 
by   1,  each  angle,  of  an  equilateral   triangle,  will   be   expressed 

Cor.  6.  In  any  triangle  ABC^  the  exterior  angle  BAD 
is  equal  to  the  sum  of  the  interior  opposite  angles  B  and 
C.  For,  AE  being  parallel  to  BC^  the  part  BAE  is 
equal  to  the  angle  B^  and  the  othei  part  DAE^  is  equal 
to   the   angle     C. 


PEOPOSITION     XXVI.       THEOKEM. 

The  sum  of  the  interior  angles  of  a  polygon  is  equal  tc 
two  right  angles  taken  as  many  times  a^  the  polygon  has 
sideSy  less   two. 

Let  AB  CDE  be  any  polygon  :  tnen  will  the  sum  of  its 
interior  angles  A^  B^  C,  I),  and  E,  be  equal  to  two  right 
angles  taken  as  many  times  as  the  polygon  has  sides,  less 
two. 

From  the  vertex  of  any  angle  A^  draw 
diagonals  AC,  AB.  ^The  polygon  will  be 
divided  into  as  many  triangles,  less  two,  as 
it  has  sides,  having  the  point  A  for  a 
common  vertex,  and  for  bases,  the  sides  of 
the  polygon,  except  the  two  which  form  the 
angle  A.  It  is  evident,  also,  that  the  sum  of  the  angles  of 
these  triangles  docs  not  differ  from  the  sum  of  the  angles  of 
the  polygon  :  hence,  the  sum  of  the  angles  of  the  polygon  is 
equal  to  two  right  angles,  taken  as  many  times  as  there  are 
triangles ;  that  is,  as  many  times  as  the  polygon  has  sides, 
less  two  ;  which  icas   to  be  proved. 


i6 


GEOMETRY. 


Cor.  1.  The  sura  of  the  interior  angles  of  a  quadrilateral 
is  equal  to  two  right  angles  taken  twice  ;  that  is,  to  four 
right  angles.  If  the  angles  of  a  quadrilateral  are  equal,  each 
■will   be   a   right   angle. 

Cor.  2.  The  sum  of  the  interior  angles  of  a  pentagon  is 
erjnal  to  two  right  angles  taken  three  times  ;  that  is,  to  six 
right  angles  :  hence,  M'hen  a  pentagon  is  equiangular,  each 
angle  is  equal  to  the  fifth  part  of  six  right  angles,  or  to  | 
of  one   right    angle. 

Cor.  3.  The  sum  of  the  interior  angles  of  a  hexagon  is 
equal  to  eight  right  angles  :  hence,  in  the  equiangular 
hexagon,  each  angle  is  the  sixth  part  of  eight  right  angles, 
or   A   of  one   right   angle. 

Cor.  4.  In  any  equiangular  polygon,  any  interior  angle  is 
equal  to  twice  as  many  right  angles  as  the  figure  has  sides, 
less  four  right  angles,  divided  by  the  number  of  angles. 


PROPOSITION     XXVII. 


THEOEEM. 


The  sum   of  the    exterior  arigles   of  a   polygon    is    equal   to 

four  right  angUs. 

Let  the  sides  of  the  polygon  ABCDE 

be   prolonged,  in   the   same  order,   forming 

he  exterior  angles  a,  h,  c,  c7,  e  ;   then  will 

the  sum  of  these  exterior  angles  be  equal 

to   four   rificht   ansjles. 

For,  each   interior  angle,  together  with 
the   corresponding   exterior  angle,   is   equal 
to   two   right  angles   (P.  I.)  :    hence,  the   sum   of  all  the 
nor  and    exterior   angles    is   equal   to   two   right   angles 


inte- 
taken 


BOOK    I.  47 

as  many  times  as  the  polygon  has  sides.  But  the  sum  of 
the  interior  angles  is  equal  to  two  right  angles  taken  as 
many  times  as  the  polygon  has  sides,  less  two  :  hence,  the 
sum  of  the  exterior  angles  is  equal  to  two  right  angles  taken 
twice  ;  that  is,  equal  to  four  right  angles ;  which  was  to  be 
proved. 


PROPOSITION      XXVm.      THEOREM. 

In  any  parallelogram,   the   opposite  sides   are  equal,  each    to 

each. 

Let  AB  CD  be  a  parallelogram  :  then 
will  AB  be  equal  to  DC,  and  AD  to 
BG. 

For,   draw  the   diagonal     BD.       Then, 
because    AB    and    DC    are   parallel,   the 
angle     DBA      is    equal    to    its     alternate 

angle  BDC  (P.  XX.,  C.  2)  :  and,  because  AD  and  BC 
are  parallel,  the  angle  BDA  is  equal  to  its  alternate  angle 
DBC.  The  triangles  ABD  and  CDB,  have,  therefore, 
the  angle  DBA  equal  to  CDB,  the  angle  BDA  equal 
to  DBC,  and  the  mcluded  side  DB  common  ;  consequently, 
they  are  equal  in  all  of  their  parts :  hence,  AB  is  equal 
to    D C,    and    AD    to    BC ',    which  was  to   be  proved. 


Cor.  1.    A  diagonal  of  a  parallelogram  divides  it  into  two 
trisngles  equal  in  all  their  parts. 

Cor.  2.  Two  parallels  included  between  two  other  par 
allels,   are  equal. 

Cor.  3.  If  two  parallelograms  have  two  sides  and  the 
included  angle  of  the  one,  equal  to  two  sides  and  the  included 
angle   of  the  other,   each  to  each,  they  will  be  equal 


48  GEOMETRY. 


PROPOSITION     XXIX.      THEOREM. 

If  the   opiyosite  sides    of  a   quadrilateral  are    equal,   each    to 
eachy   the  figure  is   a  parallelogram,. 

In  the  quadrilateral  ABCD,  let  AB 
be  equal  to  i>C,  and  AB  to  BG  '. 
then   will   it   be   a   parallelogram. 

Draw   the    diagonal    BB.       Then,   the        A 
triangles    ABB    and     CBB,    will     have 

the  sides  of  the  one  equal  to  the  sides  of  the  other,  each  to 
each  ;  and  therefore,  the  triangles  will  be  equal  in  all  of  their 
parts  :  hence,  the  angle  ABB  is  equal  to  the  angle  CBB 
(P.  X.,  S.)  ;  and  consequently,  AB  is  parallel  to  BC  (P. 
XIX.,  C.  1).  The  angle  BBC  is  also  equal  to  the  angle 
BBAy  and  consequently,  BG  is  parallel  to  AB :  hence, 
the  opposite  .  sides  are  parallel,  two  and  two  ;  that  is,  the 
figure  is  a    parallelogram   (D.  28)  ;    which  was   to   he   proved. 


PROPOSITION     XXX.      THEOREM. 

If  two   sides  of  a   quadrilateral  are  equal    and  parallel,   the 
figure  is   a  parallelogram. 

In  the  quadrilateral  ABGB,  let  AB 
he  equal  and  parallel  to  BG  \  then  will 
the  figure   be   a  parallelogram. 

Draw  the  diagonal  BB.  Then,  be- 
cause    AB     and     BG     are    parallel,    the 


angle  ABB  is  equal  to  its  alternate  angle  GBB.  Now, 
the  triangles  ABB  and  GBB,  have  the  side  BG  equal 
to  AB,  by  hypothesis,  the  side  BB  common,  and  the 
included   angle    ABB     equal    to    BBG,    from  what  has   just 


BOOK    I.  49 

been  shown;  hence,  the  triangles  are  equal  in  all  their  parts 
(P.  V,)  ;  and  consequently,  the  alternate  angles  ADB  and 
DBC  are  equal.  The  sides  BC  and  AI)  are,  therefore, 
parallel,  and  the  figure  is  a  parallelogram ;  which  was  to  be 
proved. 

Cor.  If  two  points  be  taken  at  equal  distances  from  a 
given  straight  line,  and  on  the  same  side  of  it,  the  straight 
line  joining  them  will  be  parallel  to  the  given  line. 

PEOPOSITION     XXXI.       THEOEEM. 

The    diagonals    of  a   parallelogram    divide    each    other   into 
equal  parts,  or  mutually  bisect  each  other. 

Let  ABCD  be  a  parallelogram,  and 
A  C,  £D,  its  diagonals  :  then  will  A^ 
be    equal    to     UC,    and     £B    to    IJI). 

For,  the  triangles    BUG    and    AED, 
have  the  angles    UBC   and   ADE  equal 
(P.  XX.,  C.  2),  the  angles  ECB    and   DAE    equal,  and  the 
mcluded    sides    B  G    and    AD    equal :     hence,    the    triangles 
are  equal  in  all  of  their  parts  (P.  VI.)  ;   consequently,  AE    is 
equal  to    EG,    and    BE    to    ED ;    which  was  to  be  proved. 

Scholium.  In  a  rhombus,  the  sides  AB,  BG,  being 
equal,  the  triangles  AEB,  EBG,  have  the  sides  of  the 
one  equal  to  the  corresponding  sides  of  the  other  ;  they  are, 
therefore,  equal :  hence,  the  angles  AEB,  BEG,  are  equal, 
and  therefore,  the  two  diagonals  bisect  each  other  at  right 
angles. 


^.tti 


BOOK     II. 

BATIOS       AND       PROPORTIONS. 

DEFENTTrOIirS. 

1.  The  Ratio  of  one  quantity  to  another  of  tlio  same 
kind,  is  the  quotient  obtained  by  dividing  the  second  by  the 
first.  The  first  quantity  is  called  the  Antecedent,  and  the 
second,  the  Consequent. 

2.  A  Proportion  is  an  expression  of  equality  between 
two   equal  ratios.      Thus, 

B    _  D 

A    -   C 

expresses  the  fact  that  the  ratio  of  -4  to  -S  is  equal  to 
the  ratio  of  (7  to  Z>.  In  Geometry,  the  proportion  ie 
written  thus, 

A    ',    B    x:    G    '.    Dy 

and  read,    ^    is  to    ^,    as    <7    is  to    D, 

3.  A  Continued  Proportion  is  one  in  which  several 
ratios  are  successively  equal  to  each  other  ;    as, 

A    :    B    :  I    C    '.    D  IX   E     I      F  :  :   G      :      H,      &c 

4.  lliere   are  four  terms  in  every  proportion.      The  first  I 
and  second  form  the  first  couplet,  and  the  third  and  fourth, 


BOOK    II.  51 

the  second  couplet.  The  first  and  fourth  terms  are  called 
extremes;  the  second  and  third,  means,  and  the  fourth  term, 
a  fourth  proportional  to  the  other  three.  When  the  second 
term  is  equal  to  the  third,  it  is  said  to  be  a  mean  x)roportional 
between  the  extremes.  In  this  case,  there  are  but  three; 
different  quantities  in  the  proportion,  and  the  last  is  said  to 
be  a  third  proportio7ial  to  the  other  two.      Thus,  if  we  have, 

A    :    B    '.'.    B    :     C, 

5  is  a  mean  proportional  between  A  and  (7,  and  (7  is  a 
third  proportional  to   A   and   B. 

5.  Quantities  are  in  proportion  by  alternation,  when  ante- 
cedent is  compared  with  antecedent,  and  consequent  with  con- 
sequent. 

6.  Quantities  are  in  proportion  by  inversion,  when  ante- 
cedents are  made  consequents,  and  consequents,  antecedents. 

V.  Quantities  are  in  proportion  by  composition,  when  the 
sum  of  antecedent  and  consequent  is  compared  with  either 
antecedent   or   consequent. 

8.  Quantities  are  in  proportion  by  division,  when  the  dif-' 
ference  of  the  antecedent  and  consequent  is  compared  either 
with  antecedent  or  consequent. 

9.  Two  varying  quantities  are  reciprocally  or  inversely 
proportional,  when  one  is  increased  as  many  times  as  the 
other  is  diminished.  In  this  case,  their  product  is  a  fixed 
quantity,  as  xy  =  m. 

10  Equimultiples  of  two  or  more  quantities,  are  the  pro- 
ducts obtained  by  multiplying  both  by  the  same  quantity. 
Thus,    mA   and   mB,   are   equimultiples   of  A   and   B. 


52  GEOMETRY. 


PKOPOSITION     I         THEOEEM. 

If  four    quantities    are    in    proportion^    the    product    of   the 
means  will  he  equal  to  the  product  of  the  extremes. 


Assume  the   proportion, 

A    \    B    '.'.    G    \    D\    whence,     -^    =  -q 

clearing  of  fractions,   we   have. 


BG  =  AD\ 

which  was  to  he  proved. 

Cor.  If  -S  is  equal  to  (7,  there  will  be  but  three  pro- 
portional quantities  ;  in  this  case,  the  square  of  the  mean  ib 
equal  to  the  product  of  the  extremes. 

PROPOSITION     n.        THEOEEM. 

If  the  product  of  two  quantities  is  equal  to  the  product  of 
two  other  quantities^  two  of  them  may  he  made  tJio 
meanSf  and  the  other  two  the    extremes    of  a  proportion. 

If  we  have, 

AB  =  BG, 

by  changing  the  members  of  the  equation,   we  have, 

BG  =  AB; 
dividing  both   members  by    AC,     we  have, 

B  B  A  r>  r,  Tk 

-J  =  -^  ,      or     A    :    B    '.  :     C    :    I) ; 
which  was  to  be  proved. 


m 


BOOK    II.  53 


PKOPOSITION     in.        THEOREM. 

If  fov/r  quantities    are    in  proportion^   they  wiU  be    in  pro- 
portion by  alternation. 

Assume  the  proportion, 

A    '.    B    \  \    C    \    D\    whence,  ^  =  "^  * 

Multiplying  both  members  by     >^,     we  have, 

~  =  ~\     or,     A    '.    C    XX    B    '.    B', 

which  was  to  be  proved. 


PEOPOSinON     IV.        THEOREM. 

If  one  couplet  i?i  each  of  two  proportiojis  is   the  same,  the 
other  couplets  will  form  a  proportion. 

Assume  the  proportions, 

A    '.    B    '.  :    G    :     D]     whence,     ~T  =  'n '^ 

JO  fi 

and,         A    :    B    :  :    F   :     G',     whence,    -j-  =  jr- 
From   Axiom  1,  we  have, 

Yf  =  -p  \     whence,     C    :    B    :  :    F  :     G; 
which  was  to  be  proved. 

Cor.    If  the  antecedents,  in  two  proportions,  are  the  same 
tlie    consequents  will    be    proportional.       For,   the   anteoedenta 
of  the   second   couplets   may  be  made  the  consequents   of  the 
first,  by  alternation   (P.  m.). 


64  GEOMETRY. 


PEOPOSITION     V.        THEOREM. 

If  four  quajitities    are    in  2^roportion,   they  mU   be  in  pro 

portion  by  inversion. 

Assume  the  proportion, 

A    :    B    :  :    C    '.    I>\      whence,     -j-  =  -^  • 

If  we  take  the  reciprocals  of  both  members  (A.  Y),  we  have, 

•^  =  -=r- ;     whence,    B    :    A    :  :    D    '.    C  \ 
M         JJ 

which  was  to  be  proved. 


PROPOSITION     VI.        THEOREM. 

J^  four    quantities  are    in  proportion^  they  uoiU   he  in  pro- 
portion   by  composition  or  division. 


Assume  the  proportion, 

B^  _B 

A   ~   C 


A    '.    B    :  :    G    :    I>\    whence,    -^  =  -j^ 


If  we  add     1     to  both  members,  and  subtract    1     from  both 
members,   we  shall  have, 

f+,=§+l;      and,     J_i  =  §-1,        • 

whence,  by  reducing  to  a  common  denominator,  we  have, 

B  +  A_I>+C  B-A  _  D-_C_  . 

-^jr~  G~'  '         A       -       G       '     wiience, 

A  :  B+A  :  :  G  :  B+G,    and,  A  :  B-A  :  :  G  :  J)-G 

which  was  to  be  proved. 


BOOK    II.  56 


PEOPOSITION     Vn.       THEOEEM. 

Equimultiples  of  two  quantities  are  proportional  to  the  quan- 
tities themselves. 

Let    A     and    -B    be  any  two  quantities  ;    then   -j    will 

denote  their  ratio. 

K   we    multiply  both    terms   of   this    fraction    by    »»,    its 
value  will  not  be  changed  ;    and  we  shall  have, 

— 7  =  -T  :      whence,    mA    :    mB     :  :    A    :    B  i 
mA       A*  ' 

which  was  to  be  proved. 


PROPOSITION     Vm.        THEOEEM. 

Jf  four  quantities  are  in  proportion^  any  equimultiples  of 
the  first  couplet  will  he  proportional  to  any  equimultiples 
of  the  second  couplet. 

Assume  the  proportion, 

A    :    B    '.  :    G    '.    D  \     whence,    -7-  =  -7=-  • 

AG 

If  we   multiply  both   terms   of  the  first   member  by    wi,     and 
both  terms  of  the  second  member  by    w,    we  shall  have, 

mB        nD  .  ^  ^  ^  ^ 

— 7  =  — 79  ;     whence,     mA    :    mB    :  :    nG    :    nD ; 
mA        nG 

which  was  to  he  proved. 


56  GEOMETRY. 


PEOPOSmON     IX.      THEOEEM. 

If  ttoo  quantities  be  increased  or  diminished  by  like  parte 
of  each,  the  results  will  be  proportional  to  the  quantities 
themselves. 

We  have,  Prop.  Vll., 

A    :    £    :  :    mA    :    mB. 

If  we  make      m  =  1  ±  —  ,    in  which     —     is    any  fi-actiou 
we  shall  have, 


which  was   to  be  proved. 


A    :    B    :  :    A  ±^A    :    B  ±^Bi 

q  q 


PEOPOSITION    X.        THEOEEM. 

jy^  both  terms  of  the  first  couplet  of  a  proportion  be  in- 
creased or  diminished  by  like  parts  of  each  ;  and  if  both 
terms  of  the  second  couplet  be  increased  or  diminished  by 
any  other  like  parts  of  each,  the  results  will  be  iJi  pro- 
portion. 

Since  we  have.  Prop.   VUi., 

mA    :    mB    :  :    nC    :    nD ; 

if  wo  make      m  =  1  ±  —  ,      and,     n  =  I  ±—.  .        we    shall 

q  '  q" 

have, 

A±^A    :    B±^B    ::    C±^,C    :    J)±P^nr 
q  q  q'  q'      ' 

which  was  to  be  proved. 


BOOK    II. 


67 


PEOPOSITION     Xr.        THEOEEM. 

Tn  any  continued  proportion,  the  sum  of  the  antecedents  is 
to  the  sum  of  the  consequents,  as  any  antecedent  to  its 
corresponding  consequent. 

From  the  definition  of  a  continued  proportion  (D.  3), 


A    \    B    w    G    \    B    \\    E   \    F   \  '.    G 


IS,    &c., 


hence, 


B 

B 

A 

~  A  ' 

B 

D 

A 

~  G  ' 

B 

F 

A 

-  E  ' 

B 

H  ^ 

A 

~  G  ' 

whence, 
whence, 
whence, 
whence. 


&c.. 


BA^AB\ 
BG  =zAD  \ 
BE=AF; 

BG^AH-, 

&c. 


Adding  and  factoring,  we  have, 

B{A-\-C-\-E+G-\-  &c.)  =  ^(j5+Z)+i^-l-Zr+  &c.)  : 

hence,  from  Proposition  U., 

A  +  G+  E\  (9  +  &c.    :    i?  +  i)  +  F-\-  H^  &o.    -  :  A    :  B  i 


which  was  to  be  proved. 


58  GEOMETRY. 


PROPOSITION     Xn.        THEOREM. 

Tf  two  proportions  be  multiplied  together,  term  by  term,  tJie 
products  will  be  proportional. 

Assume  tte  two  proportions, 

*  B       B 

A    :    B    :  :    C    :    J) :      whence,     -r  =  -j^i 

'  AC 

and,         E   :    F   '.  \     G    :    IT;      whence,    -=  =  ^• 
Multiplying  the  equations,  member  by  member,  we  have, 
^  =  ^ ;      whence,    AE   x    BF   :  :     CG    :    BH-, 

rahich  was  to  be  proved. 


Cor.  1.  If  the  corresponding  terms  of  two  proportions 
are  equal,  each  term  of  the  resulting  proportion  will  be  the 
square  of  the  corresponding  term  in  either  of  the  given  pro- 
portions :  hence,  ^  four  quantities  are  proportional,  their 
squares  will  be  proportional. 

Cor.  2.  If  the  principle  of  the  proposition  be  extended 
to  three  or  more  proportions,  and  the  corresponding  terms 
of  each  be  supposed  equal,  it  will  foUow  that,  like  powers 
of  proportional  quantities  are  proportionals. 


BOOK     III. 

THE     CIRCLE        AND      THB      MBASUKEMBNT      OP      ANGLES 

DEFINITIONS. 

1.  A  Circle  is  a  plane  figure, 
bounded  by  a  curved  line,  every  point 
of  whicli  is  equally  distant  from  a  point 
within,  called  the  centre. 

The  bounding  line  is  called  the  cir- 
cumference. 

2.  A  Radtcts    is    a    straight    line  drawn  from   the    centre 
to  any  point   of  the   circumference. 

3.  A    Dlameteb    is    a    straight    line    drawn    through   the 
centre   and   terminating  in   the   circumference. 

All    radii    of    the    same    circle    are    equal.       All    diameters 
are  also   equal,   and  each  is   double   the  radius. 

4.  An  Aec  is   any  part   of  a  circumference. 

5.  A  Chord  is  a  straight  line  joining  the  extremities  of 
an  arc. 

Any  chord  belongs  to  two  arcs  :   the  smaller  one  is  meant, 
miless  the   contrary  is   expressed. 

6.  A  Segment  is  a  part  of  a  circle  included  between  an 
arc  and  ita  chord. 

Y.  A  Sector  is  a  part  of  a  circle  included  by  an  arc  and  the  t^vo 
radii  drawn  to  its  extremities. 


60 


GEOMETRY. 


8.  An  Inscribed  Anglb  is  an  angle 
whose  vertex  is  in  the  circumference,  and 
whose  sides  are  chords. 


9.  Au  Inscribed  Polygon  is  a  poly- 
gon whose  vertices  are  all  in  the  circum- 
ference.   The  sides  are  chords. 

10.  A  Secant  is  a  straight  line  which 
cuts  the  circumference  in  two  points. 

11.  A  Tangent  is  a  straight  line  which 
touches  the  circumference  in  one  point  only. 
This  point  is  called,  the  point  of  contact^ 
or,  the  point  of  tangency. 

12.  Two  circles  are  tangent  to 
each  other^  when  they  touch  each 
other  in  one  pomt.  This  point  is 
called,  the  point  of  contact^  or  the 
point   of  tangency. 

13.  A  Polygon  is  circumscribed  about 
a  circle^  when  all  of  its  sides  are  tangent 
to   the   circumference. 

14.  A  Circle  is  inscribed  in  a  polygon^ 
when  its  circumference  touches  aU  of  the 
eides  of  the  polygon. 

POSTULATE. 

A    circumference    can    be    described    from    any  point   as   a 
c^tre.   and   with   any  radius. 


BOOK    III. 


61 


PEOPOSITION     I.        THEOREM. 


Any  diameter  divides  the    circle^   atid  also  its  circumference^ 

into   tico  equal  parts. 

Let  AEBF  be  a  circle,  and  AB 
any  diameter  :  then  will  it  divide  the 
circle  and  its  circumference  into  two 
equal  parts. 

For,  let  AFB  be  appUed  to  AEB, 
the  diameter  AB  remaining  common  ; 
then  will  they  coincide;  otherwise  there  would  be  some  points 
in  either  one  or  the  other  of  the  curves  unequally  distant 
from  the  centre ;  which  is  impossible  (D.  1)  :  hence,  AB 
divides  tne  circle,  and  also  its  circumference,  into  two  equal 
parts  ;  which  was   to   he  proved. 


PEOPOSITION     n.       THEOREM. 


A   diameter  is  greater  than  any  other  chord. 

Let    AD    be   a  chord,   and    AB    a  diameter  through   one 
extremity,   as    A :    then  will    AB    be  greater  than    AD. 

Draw  the  radius  CD.  In  the  tri- 
angle ACD^  we  have  AD  less  than 
the  sum  o^  AC  and  CD  (B.  L,  P. 
VJJL.).  But  this  sum  is  equal  to 
AB  (D.  3)  :  hence,  AB  is  greater 
than    AD ;    which  was  to  be  proved. 


62  GEOMETRY. 


PROPOSITION"     in.      THEOREM. 

A   straight   line    cannot  meet    a    circumference  in  more  than 

two  points. 

Let  AEBF  be  a  circumference,  and 
AB  a  straight  line :  then  AB  cannot 
meet  the  circumference  in  more  than  two 
points. 

For,  suppose  that  they  could  meet  in 
three  points.  We  should  then  have  three 
equal  straight '  Hnes  drawn  from  the  same  point  to  the  same 
straight  line  ;  which  is  impossible  (B.  I.,  P.  XV.,  C.  2)  : 
hence,  AB  cannot  meet  the  circumference  in  more  than 
two  points  ;    which  was  to   he  proved. 


PROPOSITION     IT.        THEOREM. 

In  equal  circles^   equal  arcs   are  subtended  by   equal  chords  / 
and  conversely^  equal  chords  subtend  equal  area. 

1°.  In  the  equal  cir- 
cles ABB  and  EGF, 
let  the  arcs  AMD  and 
ENG  be  equal :  then 
will  the  chords  AD  and 
EG    be   equal. 

Draw  the  diameters  AB  and  EF.  If  the  semi-circle 
ADB  be  applied  to  the  semi-circle  EGF^  it  will  coincide 
with  it,  and  the  semi-circumference  ADB  will  coincide  with 
the  semi-circumference  EGF.  But  the  part  A3ID  is  equal 
to  the  part  ERG,  by  hypothesis  :  hence,  the  point  D  will 
fell    on     G\      therefore,   the    chord     AD     will    coincide   with 


BOOK    III.  63 

EQ  (A.  11),  and  is,  therefore,  equal  to  it ;  which  was  to 
he  proved. 

2".  Let  the  chords  AD  and  EG  be  equal :  then  will 
the  arcs    AMD    and    ENG    be  equal. 

Draw  the  radu  CD  and  OG.  The  triangles  A  CD 
and  EOG  have  all  the  sides  of  the  one  equal  to  the  cor- 
responding sides  of  the  other ;  they  are,  therefore,  equal  in 
all  their  parts:  hence,  the  angle  ACD  is  equal  to  EOG, 
If,  now,  the  sector  ACD  be  placed  upon  the  sector  EOG^ 
80  that  the  angle  ACD  shall  coincide  with  the  angle  EOG., 
the  sectors  will  coincide  throughout ;  and,  consequently,  the 
arcs  AMD  and  ENG  will  coincide :  hence,  they  wUl  bo 
equal ;  which  was  to  he  proved. 


PROPOSITION     V.       THEOEEM. 

In  equal  circles.,  a  greater  arc  is  suhtended  hy  a  greater 
chord ;  and  conversely^  a  greater  chord  subtends  a  greater 
arc. 

1°.    In  the  equal  circles  JR^T^^  n  ^ — >^ 

ADL    and   EGK,    let  the         ^■ 
arc   EGP    be   greater  than       j^ 
the  arc    AMD :    then  will 
the  chord    EP    be   greater 
than  the  chord    AD. 

For,  place  the  circle  EGK  upon  AEX.,  so  that  the  cen- 
tre O  shall  fall  upon  the  centre  (7,  and  the  point  E  upon 
A  ;  then,  because  the  arc  EGP  is  greater  than  AMD.,  the 
point  P  will  fall  at  some  point  H.,  beyond  jP,  and  the 
chord    EP    will  take  the   position    AH. 

Draw  the  radii  CA.,  CD.,  and  CH.  Now,  the  sides 
AC,  CH.,  of  the  triangle  ACH,  are  equal  to  the  sides 
-4(7,    CD,    of  the  triangle    ACD,    and  the   angle    ACR   is 


64 


GEOMETRY. 


greater  than  A  CD :  hence,  the  side  AH,  or  itfJ  equal  JEP, 
is  greater  than  the  side  AD  (B.  I.,  P.  IX.)  ;  which  was  to 
be  proved. 

2°.  Let  the  chord  jE'P, 
or  its  equal  AJT,  be  great- 
er than  AD  :  then  will  the 
arc  EGP,  or  its  equal 
ADH,  be  greater  than 
AMD. 

For,  if  ADH  were  equal  to  AMD,  the  chord  AH 
would  be  equal  to  the  chord  AD  (P.  lY.)  ;  which  is  con- 
trary to  the  hypothesis.  And,  if  the  arc  ADH  were  less 
than  AMD,  the  chord  AH  would  be  less  than  AD ; 
which  is  also  contrary  to  the  hypothesis.  Then,  since  the 
arc  ADH,  subtended  by  the  greater  chord,  can  neither  be 
equal  to,  nor  less  than  AMD,  it  must  be  greater  than 
AMD  ;    which  was  to  he  proved. 


PROPOSITION     VI.        THEOEEM. 

The    radius  which    is  perpendicular  to  a  chord,   bisects    that 
chord,   and  also  the  arc  subtended  by  it. 

Let  CG  be  the  radius  which  is 
perpendicular  to  the  chord  AB  : 
then  will  this  radius  bisect  the  chord 
AB,    and   also  the  arc    AGB. 

For,  draw  the  radii  CA  and  GB. 
Then,  the  right-angled  triangles  CDA 
and  GDB  will  have  the  hypothenuse 
CA  equal  to  GB,  and  the  side  CD 
common  ;  the  triangles  are,  therefore,  equal  in  all  their 
parts  :    hence,    AD    is  equal  to    DB.      Again,   because    CG 


BOOK    III.  65 

is  perpendicular  to  AJj,  at  its  middle  point,  the  chords 
GA  and  GjB  arc  equal  (B.  I.,  P.  XVI.)  ;  and  consequently, 
the  arcs  GA  and  GB  are  also  equal  (P.  IV.)  :  hence,  CG 
bisects  the  chord  A  B,  and  also  the  arc  A  GB  ;  which  was 
to  be  proved. 

Cor.     A  straight  line,  perpendicular  to  a  chord,  at  its  mid 
die  point,  passes  through   the   centre  of  the   circle. 

Scholium.  The  centre  C,  the  middle  point  B  of  the 
chord  AB^  and  the  middle  point  G  of  the  subtended  arc, 
are  points  of  the  radius  perpendicular  to  the  chord.  But 
two  points  determine  the  position  of  a  straight  line  (A.  11) : 
hence,  any  straight  line  which  passes  through  two  of  these 
points,  will  pass  through  the  third,  and  be  perpendicular  to 
the   chord. 

PEOPOSITIOlSr    YII.       THEOEEM. 

Through  any  three  points,  not  in   the  same  straight  line,  one 
circumference  may  be  made   to  pass,   and  but   07ie. 

Let  A,  B,  and  (7,  be  any  three  points,  not  in  a 
straight  hne  :  then  may  one  circumference  be  made  to  pass 
through  them,   and   but   one. 

Join  the  points  by  the  lines 
AB,  BC,  and  bisect  these  lines 
by  perpendiculars  BE  and  FG : 
then  will  these  perpendiculars 
meet  in  some  point  0.  For, 
if  they  do  not  meot,  they  are 
parallel  ;   and   if  they   are   parallel, 

the  line  ABK,  which  is  perpendicular  to  BE,  is  also  per- 
pendicular to  KG  (B.  L,  P.  XX.,  C.  1)  ;  consequently,  there 
are    two    lines    BK     and    BF,       drawn    through    the    same 

5 


66  GEOMETRY. 

point    B^     and   perpendicular  to    the    same    line    KO ;    which 
is  impossible  :   hence,   DE    and    FG    meet  in  some  point    0. 

Now,  0  is  on  a  perpendicu- 
lar to  AB  at  its  middle  point, 
it  is,  therefore,  equally  distant 
from  A  and  B  (B.  I.,  P.  XVI.). 
For  a  like  reason,  0  is  equally- 
distant  from  B  and  C.  If, 
tlierefore,  a  circumference  be  de- 
scribed from  6>  as  a  centre,  with  a  radius  equal  to  OAy 
it  wiU  pass  through    A,     B,    and     C. 

Again,  O  is  the  only  point  which  is  equally  distant  from 
Ay  B,  and  C  :  for,  BH!  contains  all  of  the  points  which 
are  equally  distant  from  A  and  B;  and  FG  all  of  the 
points  which  are  equally  distant  from  B  and  C  ;  and  con- 
sequently, their  point  of  intersection  0,  is  the  only  point 
that  is  equally  distant  from  A^  B,  and  C  :  hence,  one 
circumference  may  be  made  to  pass  through  these  points,  and 
but   one ;  which  was  to  be  proved. 

Cor.  Two  circumferences  cannot  intersect  in  more  than 
two  points  ;  for,  if  they  could  intersect  in  three  points,  there 
would  be  two  circumferences  passing  through  the  same  three 
points  ;  which  is  impossible. 


PROPOSITION     VIII.        THEOEEM. 

Tn  equal  circles,  equal  chords  are  equally  distant  from  the 
centres  /  and  of  two  unequal  chords,  the  less  is  at  the 
greater  distance  from  t/ie  centre. 

1°.  In  the  equal  circles  ACH  and  KLG,  let  the 
chords  A  O  and  KL  be  equal :  then  will  they  be  equally 
distant   from   the   centres. 


BOOK    III. 


07 


For,  let   the  circle    KLG    be  placed  upon  ACU,    so  that 
the   centre    R     shall   fall   upon   the   centre     0,     and   the   point 
A'    upMii    the    point    A  : 
then    will    the   chord    J^L  B 

coincide     with    AG      (P.  M,W  >^  /^  ^i-" 

IV,)  ;  and  consequently, 
they  will  be  equally  dis- 
tant from  the  centre  ; 
which  was  to   be  proved. 


2".  Let  AB  be  less  than  KL  :  then  will  it  be  at  a 
greater  distance   from  the   centre. 

For,  place  the  circle  KLG  upon  AGH^  so  that  R 
shall  fall  upon  0,  and  K  upon  A.  Then,  because  the 
chord  KL  is  greater  than  AB^  the  arc  KSL  is  greater 
than  AMB ;  and  consequently,  the  point  L  will  fall  at  a 
point  C,  beyond  B^  and  the  chord  KL  will  take  the 
direction    A  G. 

Draw  OD  and  OE^  respectively  perpendicular  to  AG 
and  AB ;  then  will  OE  be  greater  than  OF  (A.  8),  and 
OF  than  OD  (B.  I,  P.  XV.)  :  hence,  OE  is  greater  than 
OB.  But,  OE  and  OD  are  the  distances  of  the  two 
chords  from  the  centre  (B.  I.,  P.  XV.,  C.  1)  :  hence,  the  less 
chord  is  at  the  greater  distance  from  the  centre ;  which  was 
to  be  proved. 

Scholium.  All  the  propositions  relating  to  chords  and  arcs 
of  equal  circles,  are  also  true  for  chords  and  arcs  of  one  and 
the  same  circle.  For,  any  circle  may  be  regarded  as  made 
up  of  two  equal  circles,  so  placed,  that  they  coincide'  in  all 

their   parts. 


es 


GEOMETRY. 


PROPOSITION     IX.        THEOREM. 

If  a  straujht  line  is  perpendicular  to  a  radius  at  its  outer 
extremity,  it  will  he  tangent  to  the  circle  at  that  p)oint ; 
conversely,  if  a  straight  line  is  tangent  to  a  circle  at  any 
jwint,  it  will  he  ^perpendicular  to  the  radius  draivn  to 
that  2Joint. 

1°.    Let     _BZ>     be    perpenclicular  to   the    radius     CM,      at 
A  :    then   ^vill  it  be   tangent  to  tlie   circle   at    A. 

For,  take  any  other  point  of 
ItDj  as  -E',  and  draw  CS : 
then  will  CJEJ  be  greater  than 
CA  (B.  I.,  P.  XV.)  ;  and  con- 
sequently, the  point  ^  will  lie 
without  the  circle  :  hence,  HD 
touches  the   circumference   at  the 

point   A',    it  is,  therefore,  tangent  to  it  at  that  point  (D.  11); 
which  was  to  he  proved. 


B 


E D 


O 


/ 


C 


2°.  Let  HD  be  tangent  to  the  circle  at  A  :  then  will 
it  be   perpendicular  to     CA. 

For,  let  £J  be  any  point  of  the  tangent,  except  the 
point  of  contact,  and  draw  CM  Then,  because  JBD  is  a 
tangent,  J?  lies  without  the  circle ;  and  consequently,  CM 
is  greater  than  CA  :  hence,  CA  is  shorter  than  any  other 
line  that  can  be  drawn  from  C  to  I)  J) ;  it  is,  therefore, 
perpendicular  to  BD  (B.  I.,  P.  XV.,  C.  1)  ;  ichich  was  to 
he  proved. 

Cor.  At  a  given  point  of  a  circumference,  only  one  tan- 
gent can  be  drawn.  For,  if  two  tangents  could  be  dra-wTi, 
they  would  both  be  perpendicular  to  the  same  radius  at  the 
same   point ;   which  is  impossible    (B.  I.,  P.  XIV.). 


BOOK    III. 


69 


PKOPOSITION     X.       THEOREM. 

I\oo  parallels  intercept  equal  arcs  of  a  circumference. 

There  may  be  three  cases:  both  parallels  may  be  secants; 
one  may  be  a  secant  and  the  other  a  tangent ;  or,  both 
may  be   tangents. 

1°.      Let    the   secants    AB     and    DB    be    parallel  :    then 
will  the  intercepted    arcs    MJ^    and    FQ    be  equal. 

For,  draw  the  radius  CIT 
perpendicular  to  the  chord 
MP  ;  it  will  also  be  per- 
pendicular to  ]^Q  (B.  I.,  P. 
XX.,  C.  1),  and  JI  will  be  at 
the  middle  point  of  the  arc 
3fIIP,  aud  also  of  the  arc 
NIIQ  :  hence,  J/iV,  which  is 
the  difference  of  iZiY  and  II3f, 

is   equal   to    FQ^     which  is   the   difference   of    MQ    and    MP 
(A.  3)  ;   tchich  was   to  be  proved. 

2°.    Let    the   secant    AB    and   tangent    JOB,    be  parallel" 
then   will   the   intercepted   arcs    3111    and    FH    be   equal. 

For,  draw  the  radius  CU 
to  the  point  of  contact  JI ; 
it  will  be  perpendicular  to  FB 
(P.  LX.),  and  also  to  its  par- 
allel 3IF.  But,  because  CU 
is  perpendicular  to  ilTP,  -S" 
is  the  middle  point  of  the  arc 
MITF  ( P.  VI.)  :  hence,  3IJI 
and  FH  are  equal ;  which 
was  to  be  proved. 


70 


GEOMETRY. 


3°.  Let  the  tangents  DE  and  IL  be  parallel,  and  let 
H  and  K  be  their  points  of  contact :  then  will  the  in- 
tercepted arcs    HMK    and    HPK   be  equal. 

For,  draw  the  secant  AB 
parallel  to  I>E ;  then,  from 
what  has  just  been  shown,  we 
ehall  have  HM  equal  to  JTP, 
and  MK  equal  to  FK:  hence, 
HMK^  which  is  the  sum  of 
HM  and  MK^  is  equal  to 
HPK^  which  is  the  sum  of 
nP  and  PK\  which  was  to 
be  proved. 


PKOPOSITION     XI.        THEOREM. 

If  two  circumferences  intersect  each  other^  the  points  of  in- 
tersection  will  he  in  a  perpendicular  to  the  straight  line 
joining  their  centres,  and  at  equal  distances  from  it. 

Let    the    circumferences,    whose    centies    are    C     and    2>, 
intersect  at  the    points    A     and 
B :   then   will     CD    be   perpen- 
dicular to    AB^    and    AF  will 
be   equal   to    BF. 

For,  the  points  A  and  B, 
being  on  the  circumference 
whose  centre  is  C,  are  equally- 
distant   from     C  ;    and   being  on 

the  circumference  whose  centre  is  i>,  they  are  equally  difr 
tant  from  D :  hence,  CD  is  perpendicular  to  AB  at  its 
middle   point   (B.  I,    P.  XVI.,  C.)  ;    which  was   to   be  proved. 


BOOK    III. 


71 


PROPOSITION     XII. 


THEOREM. 


Jf  two  circumferences  intersect  each  otJier,  the  distance  be- 
tween their  ceiitres  xcill  be  less  than  the  sum^  and  greater 
than  the  difference,  of  their  radii. 

Let    the    circumferences,    whose    centres    are     C    and    J), 
intersect  at    A  :   then   will     CD 
be     less     than      the      sum,     and 
greater    than    the    difference    of 
the   radii   of  the   two   circles. 

For,  draAV  AC  and  AD, 
foi'minjc  the  triano-le  A  CD. 
Then  will  CD  he  less  than 
the     sum     of    AC      and     AD, 

and  greater  than  their  difference    (B.  I.,  P.  VII.)  ;    which  was 
to  be  proved. 


PROPOSITION     XIII.        THEOREM. 

If  the   distance    between    the    centres    of    tico    circles  is  equal 
to  the  sum  of  their  radii,  they  will  be  tanrjent  externally. 

Let  G  and  D  be  the  centres  of  two  circles,  and  let 
the  distance  between  the  centres  be  equal  to  the  sum  of  the 
radii  :    tlien    will    the   circles   be   tangent   externally. 

For,  they  will  have  a  point 
A,  on  the  line  CD,  common, 
and  they  will  have  no  other 
point  in  common  ;  for,  if  they 
had  two  })oint3  in  common,  the 
distance  between  their  centres 
would  be  less  than  the  sum  of 
their  radii  ;  which  is  contrary  to  the  hypothesis  :  hence,  they 
are  tangent   externally  ;    which  was   to   be  proved. 


72  GEOMETRY 


PEOPOSITION     XIV.        THEOEEM. 

If  the  distance  between  the  centres  of  tioo  circles  is  equal  to 
the  difference  of  their  radii,  one  will  be  tangent  to  the 
other   internally. 

Let  C  and  D  be  the  centres  of  two  circles,  and  let 
the  distance  between  these  centres  be  equal  to  tlie  diflereuce 
of  the  radii  :  then  will  the  one  be  tangent  to  the  other  in- 
ternally. 

For,  they  will  have  a  point  A,  on 
DC,  common,  and  they  will  have  no 
other  point  in  common.  For,  if  they 
had  two  points  in  common,  the  distance 
between  their  centres  would  be  greater 
than  the  difference  of  their  radii  ; 
which    is    contrary   to    the   hypothesis : 

hence,   one    touches    the    other    internally ;    which    was    to    be 
proved. 

Cor.  1.  If  two  circles  are  tangent,  either  externally  or 
internally,  the  point  of  contact  will  be  on  the  straight  line 
drawn   through   their   centres. 

Cor.  2.  All  circles  whose  centres  are  on  the  same  straight 
line,  and  which  pass  through  a  common  point  of  that  lino, 
are  tangent  to  each  other  at  tliat  point.  And  if  a  straight 
line  be  drawn  tangent  to  one  of  the  circles  at  their  common 
point,  it   Avill   be   tangent   to   them   all   at   that   point. 

SchoUum.  From  the  preceding  propositions,  we  infer  that 
two  circles  may  have  any  one  of  six  positions  with  rcs]>ect 
to  each  other,  depending  upon  the  distance  between  ilieir 
centres  : 

1**.     When   the   distance  between    their    centres    is    greater 


BOOK    III. 


73 


tha-n    the    sum    of  their    radii,    they   are    external^    one    to    the 
other: 

2°.    "When   this   distance   is   equal   to   the   sum  of  the  radii, 
they  are  tangent,  externally: 

3°.     When   this   distance   is  less  than   the   sum,   and  greater 
ihan  the  difference  of  the  radii,  they  hUersect  each  other : 

4°.     When   this   distance  is  equal   to   the   difference  of  theii 
radii,  one  is  tangent  to  the  other,  internally: 

5°.     When   this   distance   is  less   than   the   difference    of  the 
radii,  one  is  wholly  within  the  other: 

6°.     When    this    distance     is    equal    to    zero,    they    have    a 
common  centre;    or,    they   are  concentric. 


PROrOSITIOI^"     XV.        THEOREM. 

In  equal  circles,  radii  making  equal  angles  at  the  centre, 
intercejyt  equal  arcs  of  the  circumference  ;  conversely., 
radii  xchich  intercept  equal  arcs,  make  equal  angles  at  the 
centre. 

V.  In  the  equal  circles  ABli  and  EGF,  lot  the  an- 
gles  ACD  and  EOG  be  equal:  then  will  the  arcs  AMD 
and    ENG    be   equal. 

For,  draw  the  chords  AD 
and  EG  ;  then  will  the  tri- 
angles ACD  and  EOG  have 
wo  sides  and  their  included 
angle,  in  the  one,  equal  to 
two  sides  and  their  included 
angle,  in  the  other,  each  to  each.  They  are,  therefore,  ecjual 
in  all  their  parts  ;  consequently,  AD  is  equal  to  EG. 
But,  if  the  chords  AD  and  EG  are  equal,  the  arcs  AMD 
and    ENG    are  also  equal    (P.  IV.)  ;    which  was  to  he  j^oved. 


74 


GEOMETRY. 


2°.    Let  the  arcs    AMI)    and    ENG    be  equal :   then  will 
the   angles    ACD    and    EOG    be   equal. 

For,  if  the  arcs  AMD 
and  ENG  are  equal,  the 
chords  AD  and  EG  are 
equal  (P.  IV.)  ;  consequently, 
the  triandes  ACD  and  EOG 
have  their  sides  equal,  each 
to    each  ;    they   are,   therefore, 

equal    in    all    their  parts :    hence,   the    angle    A  CD    is   equal 
to  the   angle    EOG  ;    which  was   to   he  proved. 


PROPOSITION     XVI.        THEOREM. 

In  equal  circles^  commensurable  arigles  at   the  centre  are  pro- 
portiotial  to   their  intercepted  arcs. 

In   the    equal   circles,   -whose    centres   are     C     and  0,     let 

the    angles    A  CD     and     D  OE     be    commensurable  ;  that   is, 

be   exactly   measured  by  a   common  unit:    then   will  they  be 
proportional   to    the   intercepted   arcs   AB  and  DF. 


P     7 


J-     '/. 


Let  the  angle  M  he  a,  common  unit  ;  and  suppos*.',  for 
example,  that  this  unit  is  contained  1  times  in  the  angle 
ACD,  and  4  times  in  the  angle  DOE.  Then,  supjioso 
ACD  be  divided  into  7  angles,  by  the  radii  Cm,  Cn,  Cp, 
&c.  ;  and  DOE  into  4  angles,  by  the  radu  Ox,  Oy,  and 
Oz,     each   equal   to   the   unit    M. 


V 


BOOK    III. 


76 


From  the  last  proposition,  tlie  arcs  Am^  mn^  &c.,  Dx^ 
By,  &c.,  are  equal  to  each  other  ;  and  because  there  are  1 
of  these   arcs  in  AJ3,     and-  4   in    DIJ,     we   shall   have, 


arc    AB 


arc    DU 


4. 


But,  by   hypothesis,   we  have, 


angle    ACB     :     angle    DOJS    ::     7:4; 

hence,  from   (B.  11.,  P.  IV.),   we  have, 

angle    ACJ3     :     angle    DOJS     :  :      arc    AJ3     :     arc    DX 

K  any  other  numbers  than  1  and  4  had  been  used,  the 
same  proportion  would  have  been  found ;  which  was  to  be 
proved. 

Cor.  If  the  intercepted  arcs  are  commensurable,  they  will 
be  proportional  to  the  corresponding  angles  at  the  centre, 
as  may  be  shown  by  changing  the  order  of  the  couplets  in 
the   above   proportion. 


PEOPOSITION     XVII. 


THEOEEM. 


Jn    equal    circles,    incommensurahle    angles    at    the    centre    art 
2}ro2Joriio7ial  to   their   intercepted   arcs. 

In  the  equal  circles,  whose 
centres  are  C  and  O,  let 
A  CI]  and  rOU  be  incom- 
mensurable  :  then  will  they 
be  proportional  to  the  arcs 
AB    and    FIT. 

For,  let  the  less  angle    FOII^    be  placed  upon  the  greater 
angle     A  CB^      so     that    it    shall     take     the    position     A  CD, 


dU) 


76 


GEOMETRY. 


Then,  it  the  proposition  is  not 
true,  let  us  suppose  that  the 
angle  ACB  is  to  the  angle 
FOIT,  or  its  equal  ACD, 
as  the  arc  AB  is  to  an  arc 
A  0,  greater  than  FIT,  or 
its   equal    AD ;     whence, 


angle    A  CB 


angle    A  CD 


dTc> 


arc    AB 


arc    A  0, 


Conceive  the  arc  AB  to  he  divided  into  equal  parts, 
each  less  than  DO  :  there  will  he  at  least  one  point  of 
division  between  D  and  0  ;  let  i"  be  that  point ;  and 
draw  CI.  Tlien  the  arcs  AB^  AI^  will  be  commeusurar 
ble,   and   we   shall   have    (P.  XVI.), 


angle    ACB     :     angle    ACZ 


arc    AB 


arc    AI. 


Comparing  the  two  proportions,  we  see  that  the  antecedents 
are  the  same  in  both  :  hence,  the  consequents  are  propor- 
tional   (B.  n.,  P.  IV.,  C.)  ;    hence, 


an  trie    A  CD 


an 


irle    A  CI 


arc    A  0 


arc    AI. 


But,  AO  is  gi-eater  than  AI :  hence,  if  this  proportion  is 
true,  the  angle  A  CD  must  be  greater  than  the  angle  A  CI. 
Ou  the  contrary,  it  is  less:  hence,  the  fourth  term  of  the 
assumed   proportion    cannot   be   greater   than    AD. 

In  a  similar  manner,  it  may  be  shown  that  t}ie  fourth 
terra  cannot  be  loss  than  AD  :  hence,  it  must  be  e<]ual  to 
AD  ;   therefore,  we   have, 


an 


£rle    ACB 


an 


jrie    A  CD 


arc 


AB     •      arc    AD 


which  was   to  be  proved. 


Cor.    1.     The   intercepted  arcs  are  propcrtional  to  the  cor- 


BOOK    III. 


77 


responding  angles   at   the   centre,  as  may  be   shown  by  change 
uig   the    order   of  the   couplets   in    the    in-eceding   proportion. 

Cor.  2.  In  equal  circles,  angles  at  the  centre  are  pro- 
portional to  their  intercepted  arcs  ;  and  the  reverse,  whether 
they  are   commensurable   or  incommensurable. 

Cor  3.  In  equal  circles,  sectors  are  proportional  to  their 
angles,   and   also  to   their   arcs. 

Scholium.  Since  the  intercepted  arcs  are  proportional  to 
the  corresponding  angles  at  the  centre,  the  arcs  may  be 
taken  as  the  measures  of  the  angles.  That  is,  \S  a  circum- 
ference be  described  from  the  vertex  of  any  angle,  as  a  cen- 
tre, and  with  a  fixed  radius,  the  arc  intercepted  between  the 
sides  of  the  angle  may  be  taken  as  the  measure  of  the 
angle.  In  Geometry,  the  right  angle  which  is  measured  by 
a  quarter  of  a  circumference,  or  a  quadrant,  is  taken  as  a 
unit.  If,  therefore,  any  angle  be  measured  by  one-half  or 
two-thirds  of  a  quadrant,  it  will  be  equal  to  one-half  or 
two-thirds   of  a   right   angle. 


PEOPOSITION      XVin.       THEOREM. 

An  inscribed  angle  is  measured  by  half  of  the  arc  i7icluded 

between   its   sides. 

There  may  be   three   cases  :    the   centre   of  the   circle   may 
lie  on  one  of  the  sides  of  the  angle  ;  it 
may  lie  -within  the   angle ;    or,    it    may 
lie   without  the   ansrle. 

1°.  Let  EAD  be  an  inscribed  an- 
gle, one  of  whose  sides  AE  passes 
through  the  centre  :  then  will  it  be 
measured  by  half  of  the   arc    DE. 


78 


GEOMETRY. 


For,   draw    the    radius    CD.      The    external    angle    DCE^ 
of  the   triangle    DCA,     is   equal   to   the   sum   of   the   opposite 
interior    angles    CAB     and     CD  A      (B    L,   P.  XXV.,    C.  6). 
l>ut,  the  triangle    DCA     being   isosceles, 
the    angles     I>       and     A      are     equal  ; 
therefore,    the     angle    DCE    is    double 
Ihe   angle     DAE.       Because     DCE    ia 
at  the    centre,    it    is    measured    by  the 
arc    DE     (P.  XVII.,    S.)  :      hence,   the, 
angle    DAE     is    measured    by    half   of 
the   arc    DE  ;   which  was  to  he  proved. 


2°.  Let  DAB  be  an  inscribed  angle,  and  let  the  centre 
lie  within  it :  then  will  the  angle  be  measured  by  half  of 
the   arc    BED. 

For,  draw  the  diameter  AE.  Then,  from  what  has  just 
been  proved,  the  angle  DAE  is  measured  by  half  of  DE^ 
and  the  angle  EAB  by  half  of  EB  :  hence,  BAD^  which 
is  the  sum  of  EAB  and  DAE^  is  measured  by  half  of 
the  sum  of  DE  and  EB,  or  by  half  of  BED  ;  which 
was  to  be   proved. 

3®.  Let  BAD  be  an  inscribed  angle,  and  let  the  centre 
lie  without  it  :  then  will  it  be  measured  by  half  of  the  arc 
BD. 

For,  draw  the  diameter  AE.  Then, 
from  what  precedes,  the  angle  DAE 
is  measured  by  half  of  DE,  and  the 
angle  BAE  by  half  of  BE :  hence, 
BJBy  which  is  the  difference  of  BAE 
and  DAE,  is  measured  by  half  of  the 
difference  of  BE  and  DE,  or  by 
half  of  the  arc    BD  ;    which  was  to  be  proved. 


D  E 


BOOK    III. 


79 


Cor.  1.  All  the  angles  BAG., 
BDCy  BJECf  inscribed  in  the  same 
segment,  are  equal ;  because  they  are 
each  measured  by  half  of  the  same 
arc    BOG. 


Cor.  2.  Any  angle  BAJD^  in- 
scribed in  a  semi-circle,  is  a  right  an- 
gle ;  because  it  is  measured  by  half 
the  semi-circumference  BOD^  or  by 
a  quadrant   (P.  XVII.,  S.). 


Cor.  3.  Any  angle  BAG^  in- 
scribed in  a  segment  greater  than  a 
semi-circle,  is  acute  ;  for  it  is  mea- 
sured by  half  the  arc  BOG^  less 
than   a   semi-circumference. 

Any   angle    BOG^     inscribed  in    a 
segment    less    than     a     semi-circle,    is 
obtuse  ;  for  it  is  measured    by  half  the    arc    BA  (7,    greater 
than  a  semi-circumference. 


Cor.  4.  The  opposite  angles  A 
and  C,  of  an  inscribed  quadrilateral 
ABC  J),  are  together  equal  to  two 
right  angles  ;  for  the  angle  DAB 
is  measured  by  half  the  arc  DCB^ 
the    angle    BGB     by    half   the    arc 

DAB  :  hence,  the  two  angles,  taken  together,  are  mea- 
Bured  by  half  the  circumference :  hence,  their  sum  is  equal 
to  two  right  angles. 


80 


GEOMETRY. 


PKOPOSITION     XIX. 


THEOREM. 


Any   angle  formed    by   two    chords^    which  intersect^   is   mea^ 
sured  by  half  the  sum   of  the  included  arcs. 

liCt  DEB  be  an  angle  formed  by  the  intersection  of 
the  cliords  AB  and  CD  :  then  will  it  be  measured  by 
half  the   sum   of   the   ares    A  C    and    DB. 

For,  draw  AF  parallel  to  DC: 
then,  the  arc  DF  will  be  equal  to 
AG  (P.  X.),  and  the  angle  FAB 
equal  to  the  angle  DEB  (B.  I.,  P. 
XX.,  C.  3).  But  the  angle  FAB  is  • 
measured  by  half  the  arc  FDB  (P. 
XVIII.) ;   therefore,   DEB    is  measured 

by  half  of  FDB ;  that  is,  by  half  the  sum  of  FD  and 
DB^  or  by  half  the  sum  of  A  C  and  DB ;  which  was  to 
be  "proved. 


< 


PROPOSITION      XX. 


THEOREM. 


The  angle  formed  hy  two  secants,  intersecti^ig  ivithout  the  circum- 
ference, is  measured  by  half  the  difference  of  the  included  arcs. 

Let     AB,    AC,     be    two    secants  :    then  will    the    angle 
BAC    be   measured   by  half  the  differ- 
ence  of   the   arcs    BC     and    DF. 

Draw  DE  parallel  to  AC  :  the 
arc  EC  Avill  be  equal  to  DF  (P.  X.), 
and  the  angle  BDE  equal  to  the  an- 
gle BA  C  (B.  L,  P.  XX.,  C.  3.).  But 
BDE  is  measured  by  half  the  arc 
BE  (P.  XVni.)  :  hence,  BAC  is 
also  measured  by  half  the  arc  BE ; 
that  is,  by  half  the  difference  oi  BC 
and  EC,  or  by  half  the  difference  of  BC  and  DF',  which 
war,   to   be  proved. 


BOOK    III. 


81 


PROPOSITION     XXI.        THEOREM. 


An  angle  formed  by  a  tangent  atid  a  chord  meeting  it  at 
the  point  of  contact^  is  measured  by  half  the  included 
arc. 

Let    BE    be  tangent  to  the  circle    AMC^     and  let    AC 
be   a  chord   dra\m      from       the    point    of   contact    A  :    then 
\rill    the    angle    BAG     be    measured 
by  half  of  the   arc    A3fG. 

For,  draw  the  diameter  AD. 
The  angle  BAD  is  a  right  angle 
(P.  IX.),  and  is  measured  by  half 
the  pemi-circumfcrence  AMD  (P. 
XVn.,  S.)  ;  the  angle  DAO  is 
measured  by  half  of  the  arc  DC 
(P.  XVIII.)  :   hence,  the  angle   BAC, 

which  is  equal  to  the  sura  of  the  angles  BAD  and  DAC, 
is  measured  by  half  the  sum  of  the  arcs  AMD  and  X>(7, 
or  by  half  of  the  arc    AMC  ;    which  was   to  be  proved. 

The  angle  CAB,  which  is  the  diflference  of  DAB  and  DA  0 
is  measured  by  half  the  difference  of  the  arcs  DC  A  and  DC 
or  by  half  the   arc   CA. 


PRACTICAL    APPLICATIONS. 


PROBLEM 


I. 


To  bisect  a  given  straight  line. 

Let    AB    be   a  given  straight   line. 

From  A  and  i?,  as  centres,  Avith 
3k  radius  greater  than  one  half  of  AJB^ 
describe  arcs  intersecting  at  £J  and 
F:  join  ^  and  JP,  by  the  straight 
iine  UK  Then  will  IJF  bisect  the 
given  line  AB.  For,  B  and  F 
are  each  equally  distant  from  A  and 
J) ;  and  consequently,  the  line  FF 
bisects    AJB    (B.  I.,  P.  XVI.,  C). 


I 


A 


)<E 


c 


B 


PROBLEM        n. 

To  erect  a  perpendicular  to  a  given  straight  line,  at  a  given 

point  of  that  line. 

Let   EF  be   a   given   line,  and   let  J.  be   a  given   point  o 

that  line. 

From  A,  lay  off  the  equal 
distances  AB  and '^(7;  from 
B  and  (7,  as  centres,  with  a 
radius    greater    than    one    half 


><p 


E 


t 


■+-— 


c 


F 


BOOK    III, 


83 


of  2>C,  describe  aics  intersecting  at  D\  draw  tie  line  AD: 
then  will  AD  be  the  perpendicular  required.  For,  D  and  A 
are  each  equally  distant  from  B  and  C;  consequently,  DA  is 
perpendicular   to   ^C  at  the  given  point  A  (B.  I.,  P.  XYI.,  C). 


PROBLEM     III. 


To   draw  a  perpendicular  to    a    given    straight  line.,  from  a 
given  point  without  that  line. 

Let    BD    be  the  given  line,   and    A     the  given  point. 
From    Aj     as   a   centre,   with    a  ra- 

A 

dius  sufficiently  great,  describe  an  arc 
cutting  BD  in  two  points,  B  and 
D  ;  with  B  and  D  as  centres,  and 
a   radius  greater  than  one-half  of  BD^ 


B^ 


4. 


T 

E 


-'D 


describe  arcs  intersecting  at    E\    draw 
AE  :   then   will  AE    be  the  perpendi- 
cular   required.      For,    A     and    E     are    each    equally   distant 
from    B     and    D  :     hence,     AE     is    perpendicular    to     BD 
(B.  L,  P.  XVI.,  C). 


PROBLEM     IV. 

At  a  point   on  a  given   straight   line,   to   construct   an  angle 

equal  to  a  given  angle. 

Let     A     be    the  given  point,     AB     the    given  line,    and 
[KZ    the  given  angle. 

From  the  vertex  JE'  as  a 
centre,  with  any  radius  E^I, 
describe  the  arc  IL,  terminat- 
ing in  the  sides  of  the  angle. 
From    A    aa    a.    centre,  with   a  radius    AB, 


84 


GEOMETIir. 


describe  the  indefinite  arc  UO^',  then,  with  a  radius  equal 
to  the  cliord  iJ,  from  i?  as  a  centre,  describe  an  arc 
cutting  the  arc  JOO  in  J)  ; 
draw  AD  :  then  will  JjAJ) 
bo   equal   to   the   angle    JC 

For     the    arcs     J?Z>,     7Z, 
have     equal     radii     and     equal 

chords  :    hence,  they  are  equal   (P.  IV.)  ;    therefore,  the  angles 
BABy    IKL^    measured   by   thera,   are    also   equal   (P.  XV.). 


I   A 


PROBLEM      V. 
To  bisect  a  given  arc^   or  a  given  angle. 

1°.     Let    AEB    be   a  given   arc,   and    G    its  centre. 

Draw  the  chord  AB  ;  through  C, 
draw  CD  perpendicular  to  AB  (Prob. 
m.)  :  then  will  CD  bisect  the  arc 
AEB    (P.   VL). 

2®.     Let    ACB      be   a   given   angle. 

With  (7  as  a  centre,  and  any 
radius  CB^  describe  the  arc  BA  ; 
bisect  it  by  the  line  CD,  as  just 
explained  :     then   will    CD    bisect  the   angle    ACB. 

For,  the  arcs  AE  and  EB  are  equal,  from  what  was 
just  shown  ;  consequently,  the  angles  A  CE  and  ECB  are 
also   equal     (P.   XV.). 

Scholium.  If  each  half  of  an  arc  or  angle  be  bisected, 
the  original  arc  or  angle  will  be  divided  into  four  equal 
parts  ;  and  if  each  of  these  be  bisected,  the  original  arc  or 
angle   will   be   divided   into   eight   equal   parts  ;   and   so   on. 


BOOK    III. 


86 


PROBLEM     VL 


Tlirougli   a   given   point,   to  drato    a   straight   line  parallel   to 

a  given  straight  line. 

Let    ^    be  a  given  point,   and    BG    a  given  line. 

From  the  point  ^  as  a  centre, 
\\-ith  a  radius  AEy  greater  than  the 
shortest  distance  from  A  to  JjC^ 
describe  an  indefinite  arc  -CO  ;  from 
J?  as  a  centre,  with  the  same  ra- 
dius, describe  the  arc    AJ^ ;     lay  oflf 

UD    equal  to    AE,    and  draw   AD :    then  will   AD    be  the 
parallel   required. 

For,  drawing  A£J,  the  angles  AEl^^  EAD^  are  equal 
(P.  XV.)  ;  therefore,  the  Hues  AD^  EF  are  parallel  (B.  L, 
P.  XIX.,  C.  1.). 


PROBLEM      Vn. 


Given^    two    angles    of  a     triangle^    to     construct    the    third 

angle. 

Let    A     and    D    be   given   angles   of  a  triangle. 

Diaw  a  line  DF,  and  at  some 
point  of  it,  as  E,  consti-uct  the  an- 
gle FEII  equal  to  A,  and  IIEC 
equal  to  B.  Then,  will  CED  be 
equal    to   the   required   angle. 

For,  the  sum  of  the  three  angles  at  E  is  equal  to  two 
right  angles  (B.  I.,  P.  I.,  C.  3),  as  is  also  the  sura  of  the 
three  aiigles  of  a  triangle  (B.  I,,  P.  XXV.).  Consequently, 
the  third  ang'e  CED  must  be  equal  to  the  third  angle  of 
the   triangle. 


86  GEOMETRY. 


PEOBLEM    vin. 

OiveUf  two  sides  and  the    included   angle    of  a    triangle,    to 

construct  the  triangle. 

Let   3    and    G   denote  the  given  sides,  and   A    the  given 
angle. 

Draw  the  indefinite  line  DF, 
and  at  D  construct  an  angle 
FDE,  equal  to  the  angle  A  ;  on 
BF,  lay  off  DH  equal  to  the 
Bide  C,  and  on  DE^  lay  off 
DG  equal  to  the  side  JS ;  draw 
GH :    then  will  DGIT  be  the  required  triangle  (B.  L,  P.  V.). 


PROBLEM     IX. 

Given,   one  side  and  two  angles  of  a    triangle,   to    construct 

the  triangle. 

The  two  angles  may  be  either  both  adjacent  to  the  given 
side,  or  one  may  be  adjacent  and  the  other  opposite  to  it. 
In  the  latter  case,  construct  the  third  angle  by  Problem  VIL 
We   shall  then   have  two   angles    and   their  included   side. 

Draw   a   straight  line,  and  on  it 
lay   off    DE     equal    to    the    given  Q^        ^^^ 

Bide  ;    at    D      construct    an     angle 
equal  to    one    of   the    adjacent    an- 
gles, and   at  E  construct  an   angle  D  E 
equal   to   the  other  adjacent  angle  ; 

produce   the   sides    DF    and    EG     till  they  intersect  at    Hi 
then   will    DEH    be   the   triangle    required    (B.  I ,   P.  VI.). 


V^i' 


BOOK    III. 


87. 


PROBLEM      X. 

Qiven,   the  three   sides    of   a    triangle,    to    construct    (he    tri- 
angle. 

Let    -4,    By    and    (7,    be  the  given   sides. 

Draw  DEy  and  make  it  equal 
to  the  side  A  ;  from  J[>  as  a 
centre,  with  a  radius  equal  to  the 
side  ^,  describe  an  arc  ;  from  E 
as  a  centre,  with  a  radius  equal 
to  the   side     C,     describe    an    arc 

intersecting  the   former   at     F ;     draw    DF    and    EF :     then 
will    JDEF    be   the   triangle   required    (B.  L,   P.  X.). 

Scholium.  In  order  that  the  construction  may  be  possible, 
any  one  of  the  given  sides  must  be  less  than  the  sum  of  the 
other  two,  and  greater  than  their  difference  (B.  I.,  P.  Vll.,  S.), 


Af- 
B- 


Bh 


PROBLEM      XI. 

Given,   tioo  sides   of  a    triangle^   and  the  angle  opposite  one 
of  them,   to  construct  the  triangle. 

Let    A     and    i?    be   the    given    sides,   and     C    the    given 
angle. 

Draw  an  indefinite  line  DG, 
and  at  some  point  of  it,  as  Z), 
construct  an  angle  GDE  equal 
to  the  given  angle  ;  on  one  side 
of  this  angle  lay  off  the  distance 
DE  equal  to  the  side  B  adjacent  / 

to  the  given   angle  ;    from    E    as  '' 

a  centre,  with  a  radius  equal  to  the  side  opposite  the  given 
angle,  describe  an  aic  cutting  the  side  DG  at  G  \  draw 
EG,      Then   will    BEG    be   the   required   triangle. 


88  GEOMETRY. 

For,  the  sides  DE  and  EG  are  equal  to  the  given 
sides,  and  the  angle  Z>,  opposite  one  of  them,  is  equal  to 
the   given   angle. 

Scholmm,     When    the    side    opposite    the    given    an^lo    is 

greater   than    the    other  given     side,   there   will    be    but    one 

solution.        When    the    given    angle    is    acute,    and    the    side 

<jpposite  the   given   angle   is  less 

•Jian    the    other  given   side,    and         '^' 

greater    than    the    shortest    dis-         ^'  ' 

E 
tance    from     E    to    DG,     there  j^^^^^^^^^^ 

will     be     two     solutions,     DEG  ^,^^-^^\}^  X^ 

and      DEE.        When    the    side  ^'-v.  y'^ 

opposite      the     given     angle     is 

equal  lo  the  shortest  distance  from  E  to  DG^  the  arc 
will  be  tangent  to  DG,  the  angle  opposite  DE  will  be 
a  right  angle,  and  there  will  be  but  one  solution.  When 
the  side  opposite  the  given  angle  is  shorter  than  the  distauce 
from    E    to    DG^    there   will    be   no   solution. 

PROBLE:\r     XII. 

Given^    two    adjacent    sides    of    a    j)'^'''od^^^ogram     and    their 
included    angle^    to    construct    the  parallelogram. 

Let    A     and    B    be    the    given    sides,   and     C    the   given 
angle. 

Draw   the    line     DU^     and  v .q 

at  some  point  as   Z>,    construct  /  Tp 

the  angle   JIBE  equal   to  the  /  / 

angle    C.     Lay  off  BE   equal        T>1- X n 

to  the  side  A.  and   BE  equal  / 

Ai •      / 

to   the    side     B  ;      draw    EG         p, ,  /c 

parallel  to  BE^  and  EG  par- 
allel to    BE-     then  will     BEGE    be   the  parallelogram   re- 
onired- 


BOOK    III. 


89 


For,  the  opposite  sides  are  parallel  by  construction  ;  and 
consequently,  tlio  figure  is  a  parallelogram  (D.  28)  ;  it  ie 
also   formed   with   the   given   sides   and   given   angle. 


PROBLEM      Xin. 
To  find  the  centre  of  a  given   circumference. 

Take  any  three  points  A^ 
Mj  and  C,  on  the  circumference 
or  arc,  and  join  them  by  the 
chords  AB^  BC;  bisect  these 
chords  by  the  perpendiculars  J)£J 
and  J'Yr  :  then  will  tb*^ir  point 
of  intersection  0,  be  the  centre 
required   (P.  VII.). 

SchoUnm.     The  same  construc- 
tion  enables   us   to    pass    a    circumference    through    any   three 
points  not  in   a   straight   line.       If  the   points   are   vertices  of 
a  triangle,   the   circle   will   be   circumscribed   about   it. 


PROBLEM      XIV. 
Through  a  given  pointy  to  draw  a  tangent  to  a  given  circle. 

There  may  be  two  cases :  the  given  point  may  lie  on 
the  circumference  of  the  given  circle,  or  it  may  lie  without 
the   given   circle. 

1°.  Let  C  be  the  centre  of  the 
given  oii'cle,  and  A  a  point  on  the 
circumference,  through  which  the  tan- 
gent  is   to   be   drawn. 

Draw  the  radius  CA^  and  at  A 
draw  AD  perpendicular  to  AC',  then 
will    AD    be   the   tangent   required    (P.  IX.). 


90 


GEOMETRY. 


2'=*.  Let  C  be  the  centre  of  the  given  circle;  and  A  a 
point  without  the  circle,  through  which  the  tangent  is  to  be 
drauTi. 

Draw  the  line  A  C ;  bisect  it  at 
0,  and  from  0  as  a  centre,  with  a 
radius  OC,  describe  the  circumference 
A  BCD;  join  the  point  A  with  the 
points  of  intersection  D  and  D  : 
then  will  both  AD  and  AB  be 
tangent  to  the  given  circle,  and  there 
will   be   two   solutions. 

For,  the  angles  ABC  and  ADC 
are    right    angles    (P.   XVIIL,   C.   2)  : 

hence,  each  of  the  lines  AB  and  AD  is  perpendicular  to 
a  radius  at  its  extremity  ;  and  consequently,  they  are  tangent 
to   the   given    circle    (P.  IX.). 

Corollary.  The  right-angled  triangles  ABC  and  ADC, 
have  a  common  hypothenuse  AC,  and  the  side  BC  equal 
to  DC\  and  consequently,  they  are  equal  in  all  their  parts 
(B.  L,  P.  XVn.)  ;  hence,  AB  is  equal  to  AD,  and 
the  angle  CAB  is  equal  to  the  angle  CAD.  The  tan- 
gents are  therefore  equal,  and  the  line  AC  bisects  the 
angle  between  them. 


PROBLEM      XV. 
To  inscribe  a  circle  m  a  given   triangle. 

Let  ABC  be  the  given 
triangle. 

Bisect  the  angles  A  and 
B,  by  the  lines  AO  and 
BO,  meeting  in  the  point  0 
(Prob.  V.)  ;   from   the  pouit   0 


BOOK    III. 


91 


let  fall   the   perpeucliculars     0J9,     OE^     OF^     on   tlie   sides   of 
the  triangle  :    these   perpendiculars   will   all   be    equal. 

For,  in  the  triangles  BOD  and  BOE,  the  angles  OBE 
and  OBD  are  equal,  by  construction  ;  the  angios  GDB 
and  OEB  are  equal,  because  both  are  right  angles ;  and 
consequently,  the  angles  BOD  and  BOE  are  also  equal 
(B.  L,  P.  XXV.,  C.  2),  and  the  side  OB  is  common  ;  and 
therefore,  the  ti'iangles  are  equal  in  all  their  parts  (B.  I., 
P.  VI.)  :  hence,  OD  is  equal  to  OE.  In  like  manner,  it 
may  be   shown  that    OD    is   equal   to     OF. 

From  0  as  a  centre,  Avith  a  radius  OD^  describe  a 
circle,  and  it  will  be  the  circle  required.  For,  each  side  is 
perpendicular  to  a  radius  at  its  extremity,  and  is  therefore 
tangent   to   the   circle. 

Corollary.  The  lines  that  bisect  the  three  angles  of  a 
triangle   all  meet   in   one  point. 


PROBLEM      XVI. 

On   a  given   straiglit    line,   to   construct   a  segment   that   sliall 

co7itain   a  given  angle. 


Let    AB    be  the  given  line. 


]  'reduce    AB     towui  ds     i> ;      at     i?      construct    the   angle 
DBE    equal  to   tbe    given    angle^     draw    BO    perpendicular 


92 


GEOMETRY. 


to  BE^  and  at  the  middle  point  G^  of  AB^  draw  GO 
perpendicular  to  AB  ;  from  their  ])oint  of  intersection  0, 
as  a  centre,  with  a  radius  OB^  describe  the  arc  A  MB : 
then   will   the   segment    AMB    be   the    seguient   required. 


For,  the  angle  ABF^  equal  to  EBD^  is  measured  by 
half  of  the  arc  AKB  (P.  XXI.)  ;  and  the  inscribed  angle 
AMB  is  measui-ed  by  half  of  the  same  arc  :  hence,  the 
angle  AMB  is  equal  to  the  angle  EBB^  and  conse- 
quently, to  the  given   angle. 


BOOK   IV. 


MEASUREMENT   AND   RELATION   OF   POLYGONS. 


DEFINITIONS. 


1.  SiMTLAR  Polygons,  are  polygons  which  are  mutually 
equiangular,  and  which  have  the  sides  about  the  equal  angles, 
taken   in   the   same   order,   proportional. 

2.  In  similar  polygons,  the  parts  which  are  similarly 
placed   in   each,   are   called  homologous. 

The  corresponding  angles  are  Jiomologous  angles^  the 
corresponding  sides  are  homologous  sides,  the  corresponding 
diagonals   are  homologous   diagonals,   and   so   on. 

3.  Similar  Arcs,  Sectors,  or  Segments,  in  different  circles, 
are  those  which  correspond  to  equal  angles  at  the  centre. 

Thus,  if  the  angles  A  and  0  are 
equal,  the  arcs  BFC  and  DGJE  are 
similar,  the  sectors  lyA  C  and  D  OE 
are  similar,  and  the  segments  BFC 
and    JDGE    are  similar. 

4.  Tlie    ALTiruDE    of    a    Triangle,    is    the    perpendicular 
distance    from    the  vertex    of    either  an- 
gle to  the  opposite  side,  or  the  opposite 
side   produced. 

The  vertex  of  the  angle  from  which 
the  distance  is  measured,  is  called  the 
vertex,  of  the  triangle,  and  the  opposite 
side,  is  called  the  hose  of  the  triangle. 


/ 

V 


94  GEOMETRY. 

5.  The    Altitude    op    a    Pakallelogeam,    is    the    perpen- 
dicular   distance    between    two     opposite 

sides. 

These  sides  are  called  bases  ;  one  the 
xipper^   and  the   other,   the   lovoer  base. 

6.  The    Altitude    of    a    Trapezoid,   is  the    perpendicular 
distance   between    its  parallel   sides. 

These   sides  are  called  bases  y   one  the 
uppery   and  the  other,  the  lower  base. 


1.  The  Area  op  a  Surpacb,  is  its  numerical  value 
expressed  in  terms  of  some  other  surface  taken  as  a  unit. 
The  unit  adopted  is  a  square  described  on  the  linear  iinif, 
as  a  side. 

PROPOSITION      I.        THEOREM. 

Parallelograms  ichich  have    equal    bases   and    equal  altitudes, 

are    equal. 

Let  the  parallelograms  ABCB  and  EFGH  have  equal 
bases  and  equal  altitudes :  then  will  the  parallelograms  be 
equal. 

For,  let  them  be  so  placed 
that  their  lower  bases  shall 
coincide  ;  then,  because  they 
have  the  same  altitude,  their 
upper  bases  will  be  in  the 
same   line    Z>6r,     parallel  to    AB 

The  triangles  DAH  and  CBG,  have  the  sides  AD  and 
BC  equal,  because  they  are  opposite  sides  of  the  parallel- 
ogram A  C  (B.  I.,  P.  XXVIII.)  ;  the  sides  AH  and  BQ 
equal,  because  they  are  opposite  sides  of  the  parallelogram 
AG  ;     the    angles    DAH     and    CBG    equal,    because    their 


BOOK     IV. 


95 


sides    are     parallel     and     lie    in    the    same    direction    (B.  I., 
P.  XXIV.)  :    hence,   the   triangles   are   equal    (B.  I.,    P.  V.). 

11'  from  the  quadrilateral  ABGD^  we  take  away  the  tri- 
angle DAU^  there  will  remain  the  parallelogram  AG  \  if 
from  the  same  quadrilateral  ABGD,  we  take  away  the  triangle 
CBG^  there  will  remain  the  parallelogram  AC:  hence,  the  par- 
allelogram ^C  is  equal  to  the  parallelogram  EG  (A.  3);  which 
was  to  be  proved. 


PROPOSITION     n. 


THEOREM. 


A   triangle   is    equal    to    one-half  of  a  parallelogram  having 
an  equal  base  and  an  equal  altitude. 

Let  the  triangle  AUC,  and  the  parallelogram  ABFD^ 
have  equal  bases  and  equal  altitudes  :  then  wiU  the  triangle 
be   equal  to   one-half  of  the   parallelogram. 

For,  let  them   be    so 

placed  that  the    base    of       ^. E_F. C  C 

the  triangle  shall  coin- 
cide with  the  lower  base 
of  the  parallelogram  ; 
then,  because  they  have  equal  altitudes,  the  vertex  o.  the 
triangle  will  lie  in  the  upper  base  of  the  parallelogram,  or 
in   the  prolongation  of  that  base. 

From  J.,  draw  AE  parallel  to  J5(7,  forming  the  par- 
allelograra  ABCE.  This  parallelogram  will  be  equal  to 
'the  parallelogram  ABFD^  from  Proposition  I.  But  the 
triangle  ABC  is  equal  to  half  of  the  parallelogram  ABCE 
(B.  I.,  P.  XXVm.,  C.  1)  :  hence,  it  is  equal  to  half  of 
the  parallelogram    ABED    (A.    7)  ;    which  was  to  he  proved 

Cor.    Triangles  having   equal  bases  and  equal  altitudes  are 
equal,  for  they  are  halves  of  equal  parallelograms. 


96 


GEOMETRY. 


PROrOSITION      III.        TIIEOREJI. 

Rectangles   having    equal    altitudes^   are   proportional    to   their 

bases. 

There  may  be  two  cases :  the  bases  may  be  commensu- 
rable,   or   they   may   be    incommensurable. 

1°.  Let  ABCD  antl  IIEFK,  be  two  rectangles  whose 
altitudes  AD  and  UK  are  equal,  and  whose  bases  AB 
and  HE  are  commensurable  :  then  will  the  areas  of  the 
rectangles  be   proportional   to   their  bases. 


D 

C        K 

F 

A 

I 

i 

H 

E 

Suppose  that  AB  is  to  HE,  as  7  is  to  4.  Conceive 
AB  to  be  divided  into  7  equal  parts,  and  HE  into  4 
equal  parts,  and  at  the  points  of  division,  let  perpendiculars 
be  drawn  to  AB  and  HE.  Then  will  ABCD  be  divi- 
ded into  7,  and  HEFK  into  4  rectangles,  all  of  which  will 
be  equal,  because  they  have  equal  bases  and  equal  altitudes 
(P.  I.)  :    hence,   we   have, 

ABCD     :    HEFK    :  :     7     :     4. 

But  we   have,   by   hypothesis, 

AB     :     HE     :  :      7     :     4. 

From   these   proportions,   we   have   (B.  IT.,   P.  IV".)* 
ABCD     :    HEFK     :  :    AB     :    HE. 

Had   any   otner    numbers   than   7   and   4   been  used,   the   same 
proportion   would  have   been   found  ;  which  was   to   he  j^^oved. 


BOOK     IV.  97 

2°      Let   tlie   bases  of  tlie   rectangles  be   incommensurable: 
then    will    the    rectangles   be   proportional   to   their  bases. 

For,    place    tho   rectangle    llEFK 

upon   the  rectangle    ABCD,     so  tliat  D F  K    C 

it    shall     take     the    position     AEFD. 
Then,   if   the   rectangles   are    not   pro- 
portional  to    their    bases,   let   us    suj>  ^  E~T(JB 
pose   that 

ABCD     :     AEFD     .  \    AB     :    AO; 

in    which     AO     is    greater    than     AE.        Divide  AB     into 

equal    parts,    each    less    than     OE  ;     at    least    one  point    of 

division,  as    7,    will  fall  between    E    and     0  ;     at  this  point, 

draw  IK  perpendicular  to  AB.  Then,  because  AB  and 
AI  are  commensurable,  we  shall  have,  from  what  has  just 
been   shown, 

ABCD     :    AIKI)     :  :    AB     :    AI. 

The  above  proportions  have  their  antecedents  the  same 
in   each  ;    hence    (B.  II.,   P.  IV.,  C), 

AEFD     :    AIKD     :  :    AO     :    AI. 

The  rectangle  AEFD  is  less  than  AIKD ;  and  if  the 
above  proportion  were  true,  the  line  AO  would  be  less 
than  AI ;  whereas,  it  is  greater.  The  fourth  term  of  the 
proportion,  therefore,  cannot  be  greater  than  AE.  In  like 
manner,  it  may  be  shown  that  it  cannot  be  less  than  AE ; 
consequently,   it   must   be   equal    to    AE :     hence, 

ABCD     :    AEFD     :  :    AB  AE  \ 

which  was  to  be  proved. 

Cor.    IS  rectangles    have    equal    bases,    they    are    to    each 
other   as  their  altitudes. 

7 


98 


GEOMETRY. 


PROPOSITION     IV.        THEOREM. 

Any  two  rectangles    are    to    each   other    as    the  products  of 

their  bases  and  altitudes. 

Let    AB  CD    and    AEGF   be  two  rectangles :    then    -wil 
ABCD    be  to    AEGF,     as    AB  x  AD     is  to    AE  x  AF, 

For,    place    the    rectangles    so 
that  the  angles  DAB    and  EAF 
shall    be     opposite     or     vertical ; 
then,     produce      the      sides      CD 
and    GE     till  they  meet   in    BC. 

The  rectangles  ABCD  and 
ADHE  have  the  same  altitude 
AD  :    hence   (P.  m.), 


ABCD 


ADHE     :  :    AB     :    AE. 


The    rectangles    ADHE      and     AEGF     have    the    same 
altitude    AE :    hence, 


ADHE 


AEGF 


AD 


AF. 


Multiplying  these  proportions,  term  by  term  (B.  11.,  P. 
Xn.),  and  omitting  the  common  factor  ADHE  (B.  II., 
P.  Vn.),  we  have, 

ABCD     :    AEGF    ::    AB  x  AD     :    AE  x  AF ; 
which  zoos  to  be  proved. 

Scholium  1.  If  we  suppose  AE  and  AF,  each  to  be 
equal  to  the  linear  unit,  the  rectangle  AEGF  wiU  be  the 
superficial   unit,   and  we   shall  have. 


ABCD 


AB  xAD 


1; 


BOOK     IV. 


90 


ABCD  =  AB  X  AD  : 

hence,  tfie  area  of  a  rectangle  is  equal  to  the  product  of 
its  base  and  altitude ;  that  is,  the  number  of  superficial 
units  in  the  rectangle,  is  equal  to  the  product  of  the  number 
of  linear  units  in  its  base  by  the  number  of  linear  units  in 
its  altitude. 

Scholium  2.  The  product  of  two  lines  is  sometimes  called 
the  rectangle  of  the  lines,  because  the  product  is  equal  to 
the  area  of  a  rectangle  constructed  with  the  lines  as  sides. 


PROPOSITION     V.        THEOREM. 

The  area   of  a  parallelogram  is  equal  to  the  product  of  its 

hose  and  altitude. 


Let  ABCD  be  a  parallelogram,  AB  its  base,  and  BE 
its  altitude :  then  will  the  area  of  ABCD  be  equal  to 
AB  X  BE. 

For,  construct  the  rectangle 
ABEF^  having  the  same  base 
and  altitude  :  then  will  the  rec- 
tangle be  equal  to  the  parallelo- 
gram (P.  I.)  ;  but  the  area  of  the 
rectangle  is  equal  to  AB  x  BE'. 
hence,  the  area  of  the  parallelogram  is  also 
AB  X  BE ;    which  teas  to  be  proved. 


equal     to 


Cor.  Parallelograms  are  to  each  other  as  the  products 
of  their  bases  and  altitudes.  If  their  altitudes  are  equal, 
they  are  to  each  other  as  their  bases.  If  their  bases  are 
equal,  they  are  to  each  other  as  their  altitudes. 


100 


GEOMETRY. 


PROPOSITION     VI. 


THEOREM. 


The  area  of  a  triangle  is  equal  to  half  the  product  of  its 

base  and  altitude. 

Let    ABC     be   a   triangle,     BG     its    base,   and     AD    it 
altitude :    then    will    the    area    of   the    triangle    be    equal    to 
ii?(7  X  AD. 

For,  from  (7,  draw  CE 
parallel  to  BA^  and  from  A^ 
draw  AE  parallel  to  CB.  The 
area  of  the  parallelogram  BCEA 
tg  BG  X  AD  (P.  V.)  ;  but  the 
triangle  ABG  is  half  of  the  par- 
allelogram B  CEA  :  hence,  its  area  is  equal  to  \BG  X  AD  ; 
which  was  to  be  2yroved. 

Gor.  I.  Triangles  are  to  each  other,  as  the  products  of 
th<Bir  bases  and  altitudes  (B.  IL,  P.  VII.).  If  their  alti- 
tudes are  equal,  thoy  are  to  each  other  as  their  bases.  If 
their  bases  are  equal,  they  are  to  each  other  as  their  alti- 
tudes. 

Gor.  2.  The  area  of  a  triangle  is  equal  to  half  the  pro- 
duct  of    its   perimeter   and   the   radius  of  the   inscribed   circle. 

For,  (et  DEF  be  a  circle 
inscribed  in  the  trin-ngle  ABG. 
Draw  OD,  OE,  and  OF,  to 
the  points  ot  contact,  and  OA^ 
OB,  and  OG,  to  the  verti- 
ces. 

The  area  of  OBG  will  be 
equal  to  \0E  X  BG  ;  the 
area  of   OA  G    will  be  equal  to    i OF  x  AG  ',    and  the  area 


BOOK     IV. 


101 


of  OAB  w-ill  be  equal  to  \0D  x  AJ3  ;  and  s?nco  01), 
OE,  and  OF,  are  equal,  the  area  of  the  tri.ir.glo  AUG 
(A.  9),  wUl  be  equal  to    ^OB  {AB  +  DC  ■\-  CA). 


PROPOSITION     VII.        THEOREM. 

The  area  of  a  trapezoid  is  equal  to  the  product  of  its  alti- 
tude aJid  half  the  sum  of   its  parallel  sides. 

Let  AJjCB  be  a  trapezoid,  BE  its  altitude,  and  AB 
and  I)  C  its  parallel  sides :  then  will  its  area  be  equal  to 
DEx  {{AB  +DC). 

For,  draw  the  diagonal  A  C,  form- 
ing the  triangles  ABC  and  ACB. 
The  altitude  of  each  of  these  trian- 
gles is   equal   to    BE.      The    area    of 

ABC     is  equal  to    ^AB  x  BE    (P.        AT^E  B 

VI.)  ;    the  area  of  A  CB    is   equal    to 

i^BC  X  BE:  hence,  the  aica  of  the  trapezoid,  which  is  the 
sura  of  the  triangles,  is  equal  to  the  siun  of  \AB  x  BE 
and  \BC  X  BE,  or  to  BE  x  {{AB  +  BC)  ]  which  was 
to  be  proved. 


PROPOSITION     Yin.        THEOREM. 

The  square  described  on  the  sum  of  two  lines  is  equal  to 
the  sum  of  the  squares  described  on  the  lines,  increased 
hy  twice  the  rectautjle  of  the  lines 


Let    AB    and    BC     be   two   lines, 
and    A  C    their   sum  :    then   will 


AC'  =  AB""  +  BC  -^  2AB  X  BC. 

On     A  C,       construct    the     square 
A  CDE  ;    from    B,    draw    BII    par- 


E 
F 


H 


102 


GEOMETRY. 


E        n     D 

F 

I 

0 

A            B       C 

albl  to  4i7;.  lay  off  AF  equal  to  AB,  and  from 
J;  dx-aw  F(J  parallel  to  AC  :  then  will  IG  and  IH  he 
each   equal  to  J3C  ;    and    IJ3    and    IF,     to    A£. 

The  square  A  CDF  is  composed 
of  four  parts.  The  part  ABIF  is 
a  square  described  on  AB  ;  the  part 
IGDII  is  equal  to  a  square  described 
on  BC  \  the  part  13 CGI  is  equal 
to  the  rectangle  of  AB  and  BC  'y 
and  the  part  FIHE  is  also  equal  to 
the  rectangle  of  AB    and   BC  '.    and 

because  the  whole  is  equal  to  the  sum  of  all  its  parts  (A.  9), 
we  have, 

AC""  =  AB"  +  BC''  +  2AB  x  BC  ; 

which  was  to  he  proved. 

Cor.  If  the  lines  AB  and  BC  are  equal,  the  four 
parts  of  the  square  on  AC  will  also  be  equal :  hence,  the 
square  described  on  a  line  is  equal  to  four  times  the  square 
described  on  half  the  line. 


PEOPOSITION      IX.        THEOREM. 

The  square  described  on  the  difference  of  two  lines  is  equal 
to  the  sum  of  the  squares  described  on  the  lines^  dimirir 
ished  by  twice  the  rectangle  of  the  lines. 

,    Let    AB     and    BC    be   two   lines,    and    AC    their  differ- 
ence :   then   will 


AC 


^2 


AB-  +  BC^  -  2AB  X  BC. 


On  AB  const!  uct  the  square  ABIF  \  from  C  draw 
CG  parallel  to  BI ;  lay  off  CD  equal  to  A  C,  and 
from    D    draw    DK    parallel    and    equal  to    BA  ;    complete 


BOOK     IV. 


103 


K 


E 


G 


D 


the  square  EFLK :  then  will  EK  be  equal  to  B  C,  and 
EFLK    will  be   equal   to   the   square   oi    BC. 

The  whole  figure  ABILKE  is 
erjual  to  the  sum  of  the  squares 
described  on  AB  and  BG.  The 
part  CBTG  is  equal  to  the  rect- 
angle of  AB  and  BC  ;  the  part 
DGLK  is  also  equal  to  the  rect- 
angle  of  AB    and    BC.      If   from 

the  whole  figure  ABILKE^  the  two  parts  CBIG  and 
DGLK  be  taken,  there  will  remain  the  part  ACDE^ 
which  is   equal  to   the   square   of  -4(7  :    hence, 

AC''  =  aW  +  BC^  -  2AB  X  BG  ; 
which  was  to  be  proved. 


C     B 


PROPOSITION 


THEOREM. 


The    rectangle   contained   by  the    sum  and   difference  of  two 
lineSy  is  equal  to  the  difference  of  their  squares. 

Let    AB    and    BG    be  two   lines,  of   which    AB    is  the 
greater  :    then   will 

{AB  +  BC)  {AB  -  BG)   =  AB''  -  JW- 


G     I 


On  ABy  construct  the  square 
ABIE ;  prolong  AB,  and  make 
BK  equal   to    BG;    then  will   AK  E 

be  equal  to  AB  +  BG  ;  from 
Kf  draw  EZ  parallel  to  BIj  and 
make  it  equal  to  AC  ;  draw  BE 
parallel  to  E'A,  and  CG  parallel 
to     BI :      then     BG     is    equal    to 

BC,     and    the    figure     DIIIG     is    equal    to    the    square    on 
BC,     and    EDGE    is   equal  to    BKLII. 


H 

D 

C     B     K 


104 


GEOMETRY. 


G 


D 


]l 


iL 


If  we  add  to  the  figure    ABIFE,     the   rectangle    BKLII^ 
we   shall    have   the   rectangle    A  KLE^     which   is   equal   to   the 
the     rectangle     of    All  +   II C     and 
AH  —  lie       If  to    the    same    figure 
ABITE^      Ave      add      the      rectangle 
BGFE,      e(iual     to     BKLII,       we  E 

shall  have  the  figure  .1  lUIDGF, 
which  is  e<]ual  to  the  ditft-rence  of 
the  squares  of  AB  and  11 C.  But 
the  sums  of  equals  are  c(iu:il  (A.  2), 
hence, 

(^17?  ■\-  BC)  {Ali  -  liC)  =  AB"  -  IW  ; 

which  was   to   be  proved. 


C     B     K 


PKOPOSITION      XI.        THEOREM. 

The  square  described  on  the  hi/pot/iennse  of  a  rn/ht-ajtgled 
triangle,  is  equal  to  the  sum  of  the  squares  described  on 
the   other  tioo  sides. 

Let    ABC    be   a   triangle,    right-angled   at    A  :     then    will 
BC'  =  Alf  +  AC\ 

Construct  the  square  BG  on  the  side  BCy  the  square 
AH  on  the  side  AB,  and 
the  square  AI  on  the  side 
AC  ;  from  A  draw  AB 
perpendicular  to  BC,  and 
prolong  it  to  E '.  then  will 
I)E  he  parallel  to  BE-, 
draw    AF    and    IIC. 

In  the  triangles  IlliC 
and  A  HE,  we  have  IfB 
equal  t<»  AB,  because  they 
are  sides  of  the  same  squan* ; 


BOOK     IV.  105 

BC  equal  to  JjF^  for  the  same  reason,  and  llie  included 
angles  IlliC  and  ABF  equal,  boc.uise  each  is  equal  lo  the 
angle  AliC  plus  a  right  angle  :  hence,  the  triangles  are 
equal    in    all    their    parts     (B,  L,   P.  V.). 

Tl  e  triangle  AIIF,  and  the  rectangle  BE,  have  the 
same  base  JU'\  and  because  DE  is  the  prolongation  of 
J)A^  their  altitudes  are  equal  :  hence,  the  triangle  ABE 
is  equal  to  lialf  the  rectangle  HE  (P.  II.).  The  triangle 
HBC\  and  the  square  JjL,  have  the  same  base  JUT,  and 
because  A  (J  is  the  prolongation  of  AL  (B.  I.,  P.  IV.), 
their  altitinles  are  equal:  hence,  the  triangle  II BC  is  equal 
to  lialf  the  square  of  lill.  But,  the  triangles  .J  BE  and 
IIBC  are  eijual  :  hence,  the  rectangle  BE  is  equal  to  the 
square  xVII.  In  the  same  manner,  it  may  be  shown  that 
the  rectangle  DG  is  equal  to  the  square  AI :  hence,  the 
sum  of  the  i-ectangles  BE  and  DG,  or  the  square  BG^ 
is  equal  to  the  sum  of  the  squares  All  and  xil  \  or, 
BC     =  AB    -\- AC    ;    which  was   to   be  proved. 

Cor.  1.  The  square  of  either  side  about  the  right  angle 
is  equal  to  the  square  of  the  hypothenuse  diminished  by  the 
square   of  the   other  side  :   thus, 

AB''  =  BC^  -  AC""  ;      or,      AT'^  =  BC^  -  .IB^- 

Cor.  2.  If  from  the  vertex  of  the  right  angle,  a  per- 
pendicular he  drawn  to  the  hypothenuse,  dividing  it  into  two 
segnienta,  BD  and  DG^  the  square  of  the  hypotliemtae  wiU 
he  to  the  t^ijiiitre  of  either  of  the  other  sides,  us  the  hypo- 
thenuse  is   tu  the  segment  adjacent   to   that  side. 

For,  the  s«|uaro  BG^  is  to  the  rectangle  BE,  as  BC 
to  BD  (P.  III.)  ;  but  the  rectangle  BE  is  e(jnal  to  the 
square    All  :     hence, 

WC''     :    AB"    :  :    BC     :    BD. 


106 


GEOMETRY. 


lu  like  manner,  we  have, 


BC 


AG'     :  :    BC     :    DC, 


Cor.  3.     The    squares    of  the  sides  about  the  right  angU 
are  to  each  other  as  the    adjacent 
segments  of  the  hypothenuse.  A 

For,  by  combining  the  propor- 
tions of  the  preceding  corollary 
(B.  n.,  P.  IV.,  C),  we   have,  b 


AB^ 


AG' 


BD 


DC. 


Cor.  4.     The    square    described    on    the     diago?ial    of    a 
square  is  double  the  given  square. 

For,  the  square  of  the  diagonal  is 
equal  to  the  sura  of  the  squares  of  the 
two  sides ;  but  the  square  of  each  side 
is   equal  to  the  given   square  :    hence, 


AC    =  2AB'  ;      or,     AC"  =  2BG\ 


Cor.   5.    From  the  last  corollary,  we  have. 


AG'    :    AB 


-fl 


1  ; 


hence,  by  extracting  the   square   root   of  each  term,   we   have, 
AC     \    AB     w     ^/^     \     \  \ 

that  is,  th^  diagonal  of  a  square  is  to  the  side^  as  the 
square  root  of  two  to  one ;  consequently,  the  diagonal  a7id 
the  side  of  a    square  are  incommensurable. 


BOOK     IV. 


107 


PRorosiTioN    xir.      theokem. 


In  any  triangle^  the  square  of  a  side  opjjosite  an  acute 
angle,  is  equal  to  the  sum  of  the  squares  of  the  base  and 
tJie  other  side,  dimiiiished  by  twice  the  rectangle  of  th6 
base  and  the  distance  from  the  vertex  of  the  acute  angle 
to  the  foot  of  the  perpendicular  drawn  from  the  vertex 
of  the  opposite  angle  to  tlie  base^  or  to  the  base  produced. 


Let  ABG  be  a  triangle,  C  one 
of  its  acute  angles,  -Z?C  its  base,  and 
AD  the  perpendicular  drawn  from  A 
Xa)    BCy    or    no    produced  j    then  will 


AJi'  =  BC  +  AC  -  2J3C  X  CD. 


For,  whether  the  perpendicular  meets  the  base,  or  the 
base  produced,  we  have  DD  equal  to  the  difference  of 
BG    and     CD  :     hence     (P.   IX.), 


BD'  =  BC"  +  CD*  -  2BG  x  CD. 

Adding    AD^      to    both    members,    we 
have, 


D     B 


BD'  +  AD^  =  BG'  +  CD'  +  AD'  -  2BG  x  CD. 


But,      BD""  +  AD""  =  AB\       and       CD"  +  AD"  =  AG^  : 

hence, 

AB"  =  BG^  +  A^^  -  2BC  X  CD  ; 

xohich  was  to  be  proved. 


108  GEOMETRY. 


PROPOSITION     Xni.        TIIEOUEM. 

In  any  ohtuse-avgled  triangle^  the  square  of  the  side  opposite 
the  obtuse  angle  is  equal  to  the  stim  of  the  squares  of 
the  base  and  the  other  side^  mcreased  hy  twice  the  recU 
angle  of  the  base  and  the  distance  from  the  vertex  of  tht 
obtuse  angle  to  the  foot  of  the  perpendicular  drawn  from 
the   vertex   of  the  op2)Osite  angle   to  the   base  jyroduced. 

Let  ABC  be  an  obtuse-angled  triangle,  B  its  obtuse 
angle,  BC  its  base,  and  AD  the  perpendicular  drawTi 
from    A     to    BC    produced;    then   will 


AC^  =  BC^  +  AB^  i-  2BC  X  BD. 
For,     CD    is    the    sura    of    BC         A 
and    BD:    hence    (P.  VHI.),  iV^~^\^ 

CB"  =  BC''  +  BD"  +  IBC  X  BD.  \  \        ^""^^ 

Addinj;    AD^    to   both  members,  ^    ^ 

and   reducing,    we   have, 

TC^  =  BC''  +  AB"  -f-  2BC  X  BD\ 
which  was   to   be  proved. 

Scholium.  The  right-angled  tri.mgle  is  the  only  one  in 
which  the  sum  of  the  squares  described  on  two  sides  ia 
equal   to    the   square    described   on    the   tliird   side. 

PROPOSITION     XIV.         TIIKOllEM 

In    any    triangle,    the   sum    of   the    squares   described    on    tioo 
sides    is   equal   to   twice  the   square   of  h(Uf  the  third  side 
increased    by   twice    the    square    of    the    line    dnrian   from 
the  middle  point  of  that  side  to  the  vertex  of  the  oj^osite 
angle. 
Let    ABC   be   any  triangle,   an<l    HA    a   line   drawn   from 


BOOK     IV. 


109 


the   middle   of    the   base    BC     to   the   vertex    A  :     then   will 
Alf  +  AC''  =  2BE'  +    2E7l\ 

Draw    AD    perpendicular  to    BC)    then,  from  Proposition 
XII.,   we   have, 

AC^  =   W  +  EA^  -  2EC  X  ED,  -^ 

From  Proposition   XIII.,    we   have, 
AD'   =  DE^  +  EA"  +  2DE  x  ED. 


E  13 


Adding  those   equations,  member   to   member   (A.  2),  recollect- 
ing that    DE    is  equal  to    EC,    we  have, 

AD^  +  AV"   =   2BE''  +    2EA^  ; 
which  was   to  be  proved. 

Cor.    Let     AD  CD     be    a   parallelogram,  and    DDy    AC^ 
its  diagonals.      Then,  since  the  diagonals 
mutually     bisect     each     other    (B.  I.,  P. 
XXXI.),  we  shall  have, 


and, 


AB'  +  DC   =   2AE'  +  2BE-, 


CD'  +  DA'   =   2  CE'  +  2DE'  ; 


whence,  by  addition,  recollecting  that    AE   is  equal  to    CE, 
and    BE    to    DE,    we  have, 

AB^  +  BC''  +   CD"  +  DA"  =   iCE^  +  ^DE'' ; 

but,     iCW-    is  equal  to    JT^*,    and    ^DE'    to    BW 
(P.  Vm.,   C.)  :     hence. 


AB"^  +  BC"  -f    CD"  +  DA"-   =   AC""  ■{-  BD\ 

That  is,  the  sum  of  the  squares  of  the  sides  of  a  jjaraUelo- 
graniy   is  equal  to  the  sum  of  the  squares  of  its  diagonals. 


110 


GEOMETRY. 


PTlOPOSinON     XV.        THEOREM. 

In   any  triangle^  a  line  drawn  parallel   to    the    base  divider 
the  other  sides  proportionally. 

Let    AUC     be  a  triangle,    and    DB     a    line    parallel    to 
the  base    -SC  :    then 


AD 


DB 


AE 


EC. 


Draw    EB    and    JDC.       Then,  because 

the  triandes  AED    and  DEB    have  their 

bases    in   the    same    line    AB^     and    their 

vertices  at    the    same  point    E^    they  -will 

have  a  common  altitude :    hence,   (P.  VI., 

C.) 

AED     :    DEB     :  :    AD     :    DB. 


The  triangles  AED  and  EDO,  have  their  bases  in  the 
same  line  A  (7,  and  their  vertices  at  the  same  point  D ; 
they  have,  therefore,  a  common  altitude  ;    hence, 

AED     :    EDO     :  :      AE     :    EC. 

But  the  triangles  DEB  and  EDC  have  a  common  base 
DE,  and  their  vertices  in  the  line  B  C,  parallel  to  DE ; 
they  are,  therefore,  equal :  hence,  the  two  preceding  propor- 
tions have  a  couplet  in  each  equal ;  and  consequently,  the 
-emaining  terms  are  proportional    (B.  11.,   P.  IV.),    hence, 


AE 


EC; 


AD     :    DB     : 
which  was  to  be  proved. 

Car.  1.    "We  have,  by  composition    (B.  II.,  P.  VI.), 
AD  +  DB     '.AD     '.'.    AE  ^  EC     :    AE  ; 


BOOK     IV. 


Ill 


or,  AB     :    AD 


and,   in   like  manner, 

AB     :    DB 


AC     :    AE; 


AC     '.EC. 


Cor.  2.    If  any  number   of  parallels  be   drawn  cutting  two 
lines,   they  will   divide  the  lines  proportionally. 

For,  let  0  be  the  point  where  AB 
and  CD  meet.  In  the  triangle  OEFy 
the  line  A  C  being  parallel  to  the  base 
EF^     we   shall   have, 

OE     :    AE     '.'.     OF     '.     CF. 

In  the  triangle    OGII^     we  shall  have, 

OE     '.    EG     '.'.     OF     \    FH ; 

hence    (B.  H.,  P.  IV.,  C), 

AE     \    EG     '.'.     CF     '. 


In  Uke   manner, 

EG     :    GB     :  :    FH 
and  so  on. 


fb: 


HD  ; 


PROPOSITION       XVI.        THEOREM. 

If  a  straiglit  line  divides  two  sides  of  a  triangle  2}roportionaUy, 
it  will  be  parallel  to  the  third  side. 

Let    ABC    be  a  triangle,   and  let  BE 
divide    AB    and    AC^    so  that 

AD     :    DB     '.:    AE    '.    EC  \ 

then  will    DE   be  parallel  to    BC. 

Draw    DC     and     EB.       Then   the    tri- 


112 


GEOMETRY. 


anorles    ADE    and    DE1>     will  have  a  common  altitude  ;  and 
cousequeiitly,   we   shall    have, 

A 


ABE 


DEB 


AB 


BB. 


Tlje  triuiiulcs  ABE  and  EBC  have  also 
A  common  altitude ;  and  consequently,  we 
shall   have, 

ABE     :     EBG     :  :    AE     :    EG  \ 

but,   by   hypothesis, 

AB     :    BB     :  :    AE     :    EC  ; 

hence    (B.  II.,  P.  IV.), 

ABE     :    BEB     :  :    ABE     :    EBC. 

The  antecedents  of  this  proportion  being  equal,  the  con- 
sequents will  be  equal ;  that  is,  the  triangles  BEB  and 
EBC  are  equal.  But  these  triangles  have  a  common  base 
BE  :  hence,  their  altitudes  are  equal  (P.  VI.,  C.)  ;  that  is, 
the  points  B  and  (7,  of  the  line  BC,  are  equally  distant 
from  BE,  or  BE  prolonged  :  hence,  BC  and  BE  are 
Ijarallel    (B.  I.,  P.  XXX.,   C.)  ;    which  was  to  be  proved. 


PROPOSITION      XVII.        THEOREM. 

In  any  trianrjle,  tlie  straigld  line  which  lisecfs  the  angle  a/ 
the  vertex,  divides  the  base  into  two  segments  2^ro2}ortional 
tc  the  adjacent  sides. 

Let  AB  bisect  the  vertical  angle  A  of  the  triangle 
BAC '.  then  will  the  segments  BB  and  BC  be  proper- 
tional  to  the  adjacent  sides    BA    and    CA. 

From         C,    draw    CE    parallel  to    BA,    and  produce  it 


BOOK     IV. 


113 


E 


until  it  meets  BA  prolonged,  at  K  Then,  because  CE" 
and  DA  are  parallel,  the  angles  BAD  and  A£!C  are 
equal  (B,  I.,  P.  XX.,  C.  3)  ;  the 
angles  DAO  and  ACU  are 
also  equal  (B.  L,  P.  XX.,  C.  2). 
But,  DAD  and  DAC  are 
equal,  by  hypothesis  ;  consequent- 
ly, AUC  and  ACB  are  equal: 
hence,  the  triangle  A  CE  is 
isosceles,  AJS  being  equal  to 
AC. 

In  the  triangle    DEC,     the    line    AD    is    parallel    to  the 
base    EC  :    hence   (P.  XV.), 

BA     :    AE     w    BD     \    DC  \ 

or,  substituting    AC    for  its   equal    AE^ 

BA     '.    AC     \  \    BD     '.    DC  \ 

which  was  to  be  proved. 


PROPOSITION      XVin.        THEOREM. 
Triangles  xchich  are  mutually  equiangular^   arc  similar. 

Let  the  triangles  ABC  and  DEF  have  the  angle  A 
equal  to  the  angle  i>,  the  angle  B  to  the  angle  E^  and 
the   angle    C    to   the   angle  F :    then  will  they  be  similar. 

For,  place  the  triangle 
DEF  upon  the  triangle 
ABC,  so  that  the  angle 
E  shall  coincide  with  tlie 
angle    B       then   vrill   the 

point    F      fall     at     some  ^  H     C         E 

poin^   R,   of   BC  \    the  point   D    at  some  point    (7,   of  BA  ; 

8 


114 


GEOMETRY. 


the  side    DF   will  take   the  position    GH^     and    BGH  will 
be   equal   to    EDF. 

Since  the  angle  BUG 
is  equal  to  BGAy  GH 
will  be  parallel  to  AG 
(B.  I.,  P.  XIX.,  C.  2)  ; 
and  consequently,  we  shall 
ha^e    (P.  XV.), 


H     C 


BA 


BG 


:    BG     :    BIT; 


or,   since    BG    is  equal  to    JST),     and    BM   to    FF, 
BA     :    FB     :  :    BG     :    FF. 


In  like   manner,   it  may  be  shown  that 

BG     :    FF     :  :     GA     :    FB  I 

GA     :    FB     :  :    AB     :    BE ; 


and  also. 


hence,  the  sides  about  the  equal  angles,  taken  in  the  same 
order,  are  proportional ;  and  consequently,  the  triangles  are 
similar    (D.  1)  ;    which  was  to  he  proved. 

Gor.  If  two  triangles  have  two  angles  in  one,  equal  to 
two  angles  in  the  other,  each  to  each,  they  will  be  similar 
(B.  L,  P.  XXV.,  C.  2). 


PKOPOSITION      XIX.        THEOEEM. 

Triangles  which  have  their  corresponding  sides  proportional 

are  similar. 

In  the  triangles    ABC    and    BFFy    let   the  corresponding 
sides  be  proportional ;    that  is,  let 


BOOK     IV.  115 

BA     '.ED     '.:    BC     :    EF    :  :     CA     .    FB  -, 

then  wall   the   triancrles   be   similar. 

For,    on    BA  lay  oS    BG    equal   to    FB ;     on  BC    lay 

oflf     BU     equal  to     FF, 
and     draw     GJI.         Then, 

because    BG     is  equal  to 

ED,     and    BIT  to    FF, 
we  have, 

BA     :    BG     '. 

hence,  GH  is  parallel  to  -4  C  (P.  XVI.) ;  and  consequently, 
the  triangles  BAC  and  BGH  are  equiangular,  and  there- 
fore  similar  ;    hence, 

BQ     :    BR    '.:     CA     :    HG. 
But,   by  hypothesis, 

BC     :     FF    :;     CA     :     FD\ 

hence   (B.  IT.,   P.  IV.,    C),     we   have, 

BR     :    FF    :  :    RG     :    FD. 

But,  BR  is  equal  to  EF ;  hence,  RG  is  equal  to  FB, 
The  triangles  BRG  and  ^i^Z>  have,  therefore,  their  sides 
equal,  each  to  each,  and  consequently,  they  are  equal  in  all 
their  parts.  Now,  it  has  just  been  shown  that  BRG  and 
BCA  are  similar:  hence,  FFD  and  BCA  are  also  simi- 
lar ;    which  was  to  be  proved. 

Scholium.  In  order  that  polygons  may  be  similar,  they 
must  fulfill  two  conditions :  they  must  be  mutually/  equiurtr 
gular,  and  the  corresponding  sides  must  be  proportional.  In 
the  case  of  triangles,  either  of  these  conditions  involves  the 
other,   which   is  not  true   of  any   other   species   of  polygons. 


116  GEOMETRY. 

,    PEOPOSITION     XX.        THEOREM. 

Triangles  which  have  an  angle    in    each    equals   and  the  in- 
eluding  sides  proportional^   are  similar. 

In   the  triangles    ABC    and    DEF^    let  the  angle    B    be 
equal   to   the   angle    E ;     and   suppose   that 

BA     '.    EB     '.  '.    BG     \    EF\ 

then   will   the   triangles  be   similar. 

For,  place  the  angle  E 
upon    its     equal     B  ;      F 
will    fall   at   some    point   of 
BC,     %%    n  \     D    wiU  fall 
;at  some  point   of   BA^    as        B  H     C  E  F 

G  ;  DF  will  take  the  position  GH^  and  the  triangle 
DEF  will  coincide  with  GBH^  and  consequently,  will  be 
equal   to   it. 

But,   from   the   assumed    proportion,    and    because    BG    is 
equal  to    ED^     and    BII    to    EF     ire  have, 

BA     \    BG     '.  \    BG     \    BH  \ 

hence,     GH     is    parallel   to    AG  ;  and    consequently,    BAG 

and    BGH  are    mutually  equiangular,  and   therefore   similar.     But, 

EDF  is  equal  to  BGH  :  hence  it  is  also  similar  io  BAG \  which 
was  to    he  inoved. 

PROrOSITION      XXI.        THEOREM. 

Triangles    which    have    their  sides   parallel,   each   to  each,   07 
perpoidicular,   each   to   each,   are  similar. 

1°.  Let  the  triangles  ABG  and  DEF  have  the  side 
AB  parallel  to  BE,  BG  to  EF,  and  GA  to  FI)  :. 
then   will   they  be   similar. 


BOOK     IV. 


117 


For,   since  the   side    AS     is    parallel    to    J)^,     and    J^G 
to    BFy    the  augle    -C    is   equal  to  the  angle   B     (B.  I.,  P. 
XXIV.)  ;     in  like    manner, 
the   angle     C     is    equal   to 
the   angle    i^     and  the   an- 
gle   A    to   the    angle    J)  ; 
the  triangles  are,  therefore, 
mutually    equiangular,     and        " 
consequently,    are    similar    (P.    XVIII.)  ;      which    was    to    be 
proved. 

2°.  Let  the  triangles  AUG  and  I>EF  have  the  side 
AB  perpendicular  to  DJE,  BO  io  EF,  and  CA  to 
FD  :     then  t\tI1  they  be   similar. 

For,  prolong  the  sides  of  the  tri- 
angle DEF  till  they  meet  the  sides 
of  the  triangle  ABC.  The  sum  of 
the  interior  angles  of  the  quadrilateral 
BIEG  is  equal  to  four  right  angles 
(B.  I.,  P.  XXVI.)  ;  but,  the  angles 
EIB      and    EGB      are     each     right 

angles,  by  hypothesis ;  hence,  the  sura  of  the  angles  lEG 
IBG  is  equal  to  two  right  angles  ;  the  sum  of  the  angles* 
lEG  and  DEF  is  equal  to  two  riglit  angles,  because  the} 
are  adjacent ;  and  since  thmgs  which  are  equal  to  the  same 
thing  are  equal  to  each  other,  the  sum  of  the  angles  lEO 
and  IBG  is  equal  to  the  sum  of  the  angles  lEG  and  DEF\ 
or,  taking  away  the  common  part  lEG^  we  have  the  angle 
IBG  equal  to  the  angle  DEF.  In  like  manner,  the  angle 
GCII  may  be  proved  equal  to  the  angle  EFD^  and  the 
angle  HAI  to  the  angle  EDF  \  the  triangles  ABC  and 
DEF  are,  therefore,  mutually  equiangular,  and  consequently 
similar  ;    which   was   to   be  proved. 

Cor.  1.     In   the   first   case,   the    parallel    sides    are   horaolo- 


118 


GEOMETRY. 


gous  ;    in  the   second   case,  the  perj)cndicular  sides   are  homo- 
logons. 

Cor.  2.  The  homologous  angles  are  those  included  by 
sides  respectively  parallel   or  perpendicular  to   each   other. 

Scholium.  When  two  triangles  have  their  sides  perpcn- 
iicular,  each  to  each,  they  may  have  a  different  relative 
position  from  that  shown  in  the  figure.  But  we  can  always 
construct  a  triangle  within  the  triangle  ABC,  whose  sides 
shall  be  parallel  to  those  of  the  other  triangle,  and  then  the 
demonstration  will  be  the  same  as   above. 


P^.OPOSITION     XXn.        THEOREM. 

If  a  straight  line  le  drawn  parallel  to  the  lase  of  a  triangle, 
and  straight  lines  le  drawn  from  the  vertex  of  the  triangle 
to  points  of  the  base,  these  lines  will  divide  the  base  atid 
the  2^arallel  pro2}ortionalh/. 

Let  ABC  be  a  triangle,  BC  its  base,  A  its  vertex, 
DJE  parallel  to  BC,  and  AF,  AG,  AH,  lines  drawn 
from    A    to   points   of  the  base  :    then   wiU 

DI    '.  BF   '.'.  IK   '.  FG    '.'.  KL    :    GH   :  :  LE   :  HG, 


For,     the      triangles    AID      and 
AFB,    being   sunilar  (P.  XXL),  we 

have, 

AI    '.    AF    w    BI    '.    BF', 

and,  the  triangles  AIK  and    AFG, 
being   similar,   we   have, 

AI    :    AF    :  :    IK 


hence,    (B.  II.,   P.  IV.),  we  have, 


BOOK     IV. 


119 


BI    '.    BF    '.'.    IK 


FG. 


In  like   manner, 

IK     '.    FO     w    KL     ',     GH, 

and, 

KL    \     GH     '.  :    IE     :    no  \ 

hence     (B.  H.,  P.  IV.)> 

DI    :  BF   '.  :  IK   :  FG   :  :  KL   '.   GH   :  :  LE    \  HC  \ 

which  teas   to  be  proved. 

Cor.  If  BC  is  divided  into  equal  parts  at  F,  G,  and 
H,  then  will  BE  be  divided  into  equal  parts,  at  7",  K.^ 
And    I. 


PROPOSITION      XXIII.        THEOREM. 

7j^,  in  a  right-angled  triangle^  a  perpendicular  be  drawn  from 
the  vertex  of  the  right  angle  to  the  hypothenuse  : 

1".  The  tria7igles  on  each  side  of  the  perpendicular  will  be 
similar  to  the  given  triangle^  and  to  each  other : 

2°.  Each  side  about  the  right  angle  will  be  a  mean  propor- 
tional between  the  hypothenuse  and  the  adjacent  segment : 

3°.  The  perpendiddar  will  be  a  mean  proportional  between 
the  two  segments  of  the  hypothenuse. 

1°,  Let  ABC  be  a  right-angled  triangle,  A 
of  the  right  angle,  BC  the  hypo- 
thenuse, and  AD  perpendicular  to 
BC  :  then  will  ADB  and  ADC 
be  similar  to  ABC,  and  conse- 
quently,  similar  to   each   other. 

The  triangles   ADB    and    ABC 
have    the    angle     B      common. 


B  DC 

and    the    angles    ADB    and 


^^ 


120 


GEOMETRY. 


BA C  equal,  because  both  aie  right  angles  ;  they  are,  there 
fore,  similar  (P.  XVIII.,  C ).  In  like  manner,  it  may  be 
sliown  that  the  triangles  ADC  and  ABC  are  similar; 
and  since  ABB  and  ADC  are  both  similar  to  ABC, 
they   are   similar  to   each   other  ;   which  was   to   be  jyrovcd. 

2'^.  ^17?  will  be  a  mean  pro- 
portional between  BC  and  BD ; 
and  A  C  will  be  a  mean  propor- 
tional   between     CB    and     CD. 

For,  the  triangles  AD/>  and 
BAC  being  similar,  their  homo- 
logous  sides   are   proportionu!  :    hence. 


BG 


AB 


AB 


BD. 


In   like   manner, 

BC     :    AC 
which  was  to  be  proved. 


AG    :    DC 'y 


3°.  AD  will  be  a  mean  i»roportional  between  BD  and 
DC.  For,  the  triangles  .!/>/>'  and  ADC  being  similar, 
their   homologous  sides   are   proportional ;    hence, 


BD     :    AD 
which  was  to  be  proved. 


AD 


DC  i 


and. 


Cor.  I.     From  the   proponions, 

BG     :     AD     :  :     AB     :     BD, 
BG     :    AC     :  .     AC     :    DC, 
we  have    (B.  E.,   P.  I.), 

AB^    =    nC  X  BD, 
AC"-    =    HG  X  DC  ; 


and, 


BOOK     IV. 


121 


whence,  by   addition, 


AB'  +  AC  =  BC{BD  +DG)  ; 


or. 


AB'  +  AC  =  liC'  ; 


as  was  shewn  in  Proposition  XI. 


Cor.  2,  If  fi-om  any  point  A^  in  a  semi-circumference 
BACy  chords  be  drawn  to  the 
extremities  B  and  C  of  the  diam- 
eter J?C,  and  a  perpendicular  AD 
be  drawn  to  the  diameter  :  then 
will  AB  (7  be  a  right  sngled  tri- 
angle, right-angled  at  A  ;  and  from  what  was  proved  above, 
each  chord  will  he  a  mean  jyroportional  between  the  diameter 
and  the  adjace?it  segmeyit  /  and,  tJie  perpendicidar  will  be  a 
mean  proportional  between  the  segments  of  the  diameter. 


B    D 


PROPOSITION      XXIV.        THEOEEil. 


Triangles  which  have   an  angle  in    each  equal^   are    to    each 
other  as   the  rectangles   of  the  including   sides. 

Let  the  triangles  GSK  and  ABC  have  the  angles  G 
and  A  eqiial :  then  will  they  be  to  each  other  as  th.^ 
r.'ctangles   of  the   sides   about   these   angles. 

For,  lay  ofif  AD  equal 
to  GH,  AE  to  GK,  and 
draw  DE ;  then  will  the 
triangles  ADE  and  GHK 
be  equal  in  all  their  parts. 
Draw    EB. 


II 


■K 


122 


GEOMETRY. 


The   triangles    ABE  and    ABE    have   their  bases  in   the 

same  line    AB^    and    a  common  vertex    E ;    therefore,   they 

have  the   same    altitude,  and   consequently,   are   to   each   other 
as  their   bases  ;   that  is, 

ABE    :    ABE    :  :    AB     :    AB. 


The  triangles  ABE  and 
ABC^  have  their  bases  in 
the  same  line  AC^  and  a 
common  vertex   B  ;    hence, 


ABE    :    ABC 


AE    :    AC; 


multiplying     these     proportions,     term      by     term,      and     omitting 
the  common   factor  ABB    (B.  II.,  P.  VII.),  we   have, 

ABE    :    ABC    :  :    AB  x  AE    :    AB  x  AC', 

substituting  for  ABE,    its  equal,    GIIE,    and  for    AB  x  AE, 
its   equal,     Gil  X   GE,     we   have, 

GIIE   :    ABC    :  :     Gil  x  GK   :    AB  x  AC; 

which  was   to  be  proved. 

Cor.  If  ABE  and  ABC  are  similar,  the  angles  B 
and  B  being  homologous,  BE  will  be  parallel  to  BC, 
and  we   shall   have, 


AB 


AB 


AE 


AC  ; 


hence   (B.  IT.,   P.  IV.),    we  have, 

ABE     :    ABE     :  :    ABE     :     ABC  ;  ° 

that  is,    ABE   is  a  mean   proportional  be- 
tween   ABE    and    ABC. 


BOOK     IV. 


123 


PROPOSITION     XXV.        TUEOREM. 

Similar  triangles   are   to  each  other    as    the  squares  of  their 

homologous  sides. 

Let  the  triangles  ABC  and  DEF  be  similar,  the  angle 
A  being  equal  to  the  angle  D,  JD  to  E^  and  C  to  F. 
then  will  the  triangles  be  to  each  other  as  the  squares  of 
any  two   homologous   sides. 

Because    the    angles    A    and    D     are    equal,  we  have   (P. 

xxrv'.), 

ABC     :    DBF    ::    AB  x  AC     :    DE  x  DF \ 


and,     because      the     triangles 
are    similar,   we    have, 

AB  :  BE  :  :  AC    :  BE; 

multiplying     the      terms     of      ^ 
this    proportion    by   the    cor- 
responding  terms   of  the   proportion, 

AC     :    BE    :  :    AC     :    BE, 
we  have    (B.  H.,   P.  XII.), 


AB  X  AC    :    BE  x  BE   :  :    AC'    :    BE-, 

combining     this,  with    the     first     proportion      (B.  II.,   P.  FV.), 
we  have, 

ABC    :    BEE   :  :    AC^         BE . 

In  like  manner,  it  may  be  shown  that  the  triangles  are 
to  each  other  as  the  squares  of  AB  and  BE^  or  of  BC 
and    EF ;    which  was   to  be  proved. 


124 


GEOMETRY. 


PROPOSITION      XXVI. 


THEOREM. 


Similar  polygons  may   be  divided  into   the   same  number  of 
triangles^   similary   each,  to  each^   and  sifnilarly  placed. 

let  ABCDE  and  FGHIK  be  two  similar  polygons, 
the  angle  A  being  equal  to  the  angle  F^  B  to  G^  C  to 
21,  and  so  on  :  then  can  they  be  divided  into  the  same 
number   of  similar   triangles,   similarly   placed. 

For,   from    A    draw 
the     diagonals        AC,  G 

AD,       and     from     F,  B 

homologous  with  A , 
draw  the  diagonals 
FJI,  FT,  to  the  ver- 
tices IT  and  /,  hom- 
ologous  with     C    and    D. 

Because  the  polygons  are  similar,  the  triangles  ABC  and 
FGIT  have  tlie  angles  B  and  G  equal,  and  the  sides 
about  these  angles  proportional  ;  they  are,  therefore,  similar 
(P.  XX.).  Since  these  triangles  are  similar,  we  have  the 
angle  ACB  equal  to  FTTG,  and  the  sides  AC  and  i^-S", 
proportional  to  BC  and  GIT,  or  to  CD  and  TIT.  The 
angle  BCJJ  being  equal  to  the  angle  GITT,  if  we  take 
from  the  first  the  angle  ACB,  and  from  the  second  the 
equal  angle  FTTG,  we  shall  have  the  angle  A  CD  equal 
to  the  angle  FHT :  hence,  the  triangles  A  CD  and  FTTI 
have  an  angle  in  each  equal,  and  the  inchuling  sides  j)ropor- 
tional ;    they    are   therefore  similar 

In  like  manner,  it  may  be  shown  that  ADE  and  FTK 
are   similar  ;     which  was  to   be  proved. 

Cor.  1.  The  corresponding  triangles  in  the  two  polygons 
are  homoloyous  triangles,  and  the  corresponding  diagonals  are 
homologous   diagonals. 


BOOK    IV. 


125 


Cor.  2.    Any  two    homologous    triangles  are   like  2)a?is    of 
the   polygons   to   which   they   belong. 

For,   the  homologous  triangles  being  similar,   we  have, 


ABC  : 

FGH 

:  AC' 

:  Fit-; 

ttUd, 

ACD  : 

Fill 

:  :  AC' 

:  Fit; 

whence, 

ABC: 

FGII 

:  :  ACD 

:  FUI. 

But, 

ABC  : 

FGH 

:  ABC 

:  FGH; 

and. 

ABC  : 

FGH    • 

.  ADE 

:  FIX; 

by  composition, 

ABC  .  FGH  :  :  ACD  +  ABC+ADF:  Fill +FG 11+ FIX; 
that  is,  ABC  :  FGH  :  :  ABCDE  :   FGH  IK.    ' 

Cor.  3.     If  two  polygons  are  made  up  of  similar  triangles, 
similarly  placed,  the  polygons  themselves  will  be  similar. 


PROPOSITION     XXVII. 


THEOREM. 


The  perimeters  of  similar  polygons  are  to  each  other  as  any 
two  homologous  sides  ;  and  the  2^oIygons  are  to  each 
other  as   the   squares  of  any   two  homologous    sides. 

1".  Let  ABCDE  and  FGHIK  be  similar  polygons: 
then  will  their  perimeters  be  to  each  other  as  any  two 
homologous   sides. 

For,  any  two  horao- 
locrous  sides,  as  AB 
and  FG^  are  like  parts 
of  the  perimeters  to 
which  they  belong  : 
hence  (B.  H.,  P.  IX.), 
the   perimeters    of    the 

polygons  are   to   each    other    as     AB     to     FG., 
other  two  homologous  sides  ;    which  was  to  be  proved. 


or    as    any 


126  GEOMETRY. 

2°.    The    polygons  will    be    to    each    other  as  the   squares 
of  any  two   homologous   sides. 

For,    let    the    poly-  ^ 

ffons    be    divided     into  t>  _^-— -■'''^?^^  _-^ 

homolofirous       trianc^les  \       /'  \        [     /'         \ . 

(P.     XXVI,      C.    1)  ;  ^\/ -^^I^    "X 

then,        because        the  ^^^^/''^  ^^ 

homologous       triangles  E 

ABC    and  FGH    are  . 

like  parts  of  the  polygons  to  which  they  belong,  the  poly- 
gons will  be  to  each  other  as  these  triangles ;  but  these 
triangles,  being  similar,  are  to  each  other  as  the  squares  of 
AJB  and  FG  :  hence,  the  polygons  are  to  each  other  as 
the  squares  of  AB  and  FG^  or  as  the  squares  of  any 
other  two  homologous   sides  ;    which  was   to   be  proved. 

Cor.  1.  Perimeters  of  similar  polygons  are  to  each  other 
as  their  homologous  diagonals,  or  as  any  other  homologous 
lines  ;  and  the  polygons  are  to  each  other  as  the  sqiaares  of 
their  homologous  diagonals,  or  as  the  squares  of  any  other 
homologous   lines. 

Cor.  2.  If  the  three  sides  of  a  right-angled  triangle  be 
made  homologous  sides  of  three  similar  polygons,  these  poly- 
gons ynW.  be  to  each  other  as  the  squares  of  the  sides  of 
the  triangle.  But  the  square  of  the  hypothenuse  is  equal 
to  the  sum  of  the  squares  of  the  other  sides,  and  conse- 
quently, the  polygon  on  the  hypothenuse  will  he  equal  tc 
he  sum  of  the  polygons  on  the  other  sides. 

PEOPosiTioN    xxvrn.     theorem. 

If  two  chords    intersect    in    a    circle^   their    segments  wiU   be 

reciprocally  proportional. 

Let    the    chords    AB     and     CD     intersect  at    0  :    then 


BOOK     IV.  127 

will  their  segments  be  reciprocally  proportional  ;  that  is,  one 
segment  of  the  first  will  be  to  one  segment  of  the  second, 
as  the  remaining  segment  of  the  second  is  to  the  remaining 
segment   of  the   first. 

For,  draw  CA  and  JBD.  Then 
(vill  the  angles  ODB  and  OAC  be 
ecjual,  because  each  is  measured  by  half 
of  the  arc  CB  (B.  HI.,  P.  XVni.). 
The  angles  OBB  and  OCA,  wiU  also 
be   equal,   because    each    is    measured   by 

half  of  the  arc  AD :  hence,  the  triangles  OBD  and  0  CA 
are  similar  (P.  XVIII.,  C),  and  consequently,  their  homolo- 
gous  sides   are  proportional :    hence, 

DO    :    AO    :  :     OB    :     OC  I 

which  was  to  be  proved. 

Cor.     From   the   above   proportion,   we   have, 

DO  X  OC    =    AOx  OB  I 

that  is,  the  rectangle  of  the  segments  of  one  chord  is  equal 
to  the  rectangle  of  the  segments  of  the  other. 


PEOPOSmOlT     XXIX.        THEOEEM. 

If  from  a  point  without  a  circle,  two  secants  he  drawn  tcr- 
m,inating  in  the  concave  arc,  they  wiU  be  reoiprocally 
proportional  to  their  external  segments. 

Let    OB    and    OC     be    two    secants    terminating    in    the 
concave  arc  of  the  circle    BCD  :    then  will 

OB     i     OC     :  \     OD     :     OA, 


128 


GEOMETRY. 


For,    draw    AC     and    DJ3.        The    triangles     ODB     and 
OA  C     have    the    angle     0     common,    and    the   angles     OBD 
and    OCA    equal,  because  each   is  measured 
by   half  of  the   arc    AD  :     hence,    they   are 
isimjlar,   and    consequently,   tlieir  homologous 
sides  are  proportional ;    whence, 


OB     :     00     : :     OD 

which  was  to  be  proved. 


OA  ; 


Cor.    From    the    above    proportion,     we 

have, 

OB  X  OA    =    00  X  OD  ; 

that  is,   the  rectangles  of  each  secant    and   its    external   seg- 
ment are  equal. 


PROPOSITION  XXX.        THEOREM. 

If  from   a  point  without  a  circle^   a    tangent   and  a  secant 
be  drawn^   the  secant   terminating   in   the  concave  arc^   the 

tangent  will    be    a    mean  proportional  between  the    secant 
and  its  external  segment. 

Let    ADC     be   a  circle,  OC    a  secant,   and    OA    a  tan- 
gent :    then  will 

OC    :     OA  :  :     OA     :     OD. 


For,  draw  AD  and  AC.  The  tri- 
angles OAD  and  OAC  will  have  the 
angle  0  common,  and  the  angles  OAD 
and  A  CD  equal,  because  each  is  mea- 
sured by  half  of  the  arc  AD  (B.  HI., 
P.  XVm.,  P.  XXI.)  ;  the  triangles  are 
therefore   similar,    and   consequently,   their 


BOOK     IV.  129 

homologous   sides   are   proportional :    hence, 

OC     :     OA     :  :     OA     :     OD  \ 
which  toas  to  be  proved. 

Cor.    From   the  above  proportion,  we  have, 

.   JTO'    =    OC  X  OD  i  i 

that  is,   the  square  of  the    tangent  is  equal  to  the  rectangle 
of  the  secant  and  its  external  segment. 


PRACTICAL    APPLICATIONS. 


PROBLEM     I. 

To  divide  a  given  straight  line  into  parts  proportional  to  given 
straight  lines:  also  into  equal  parts. 

1*.    Let  AB  be  a  given  straight  line,  and  let  it  be  required 
to  divide  it  into  parts  proportional  to  the  lines  P,   Q,  and  R. 

From  one  extremity  -4, 
draw  the  indefinite  line  AGi, 
making  any  angle  T\-ith  AD  ; 
lay  oS  AC  equal  to  P,  CB 
equal  to  Q^  and  BE  equal 
to  R  ;  draw  EB^  and 
from  the  points  C  and  Z>, 
draw  CI  and  BF  parallel  to  EB :  then  will  ^/,  JF, 
and    FB,    be  proportional  to    P,    Q,    and    R   (P  XV.,  C.  2). 

9 


130 


GEOMETRY, 


2°.    Let  AH  be  a  given  straight  line,  and  let  it  be  required 
to   divide   it  into   any   nnmber    of  equal   parts,   say  five. 

From     one     extremity 

Ay      draw    the     indefinite  a^ G E E F. H 

line  A  G  ;  take  AT  equal 
to  any  convenient  line, 
and  lay  off  IJT]  KL^ 
LMy  and  MB^  each 
equal  to  AI.  Dravs^ 
BHy     and    from     J,     K,     Z, 


and     ilT, 
KB,    LE,    and    MF,    parallel   to    BH 


draw  tlie  lines    IG, 
then   will    AH   be 


divided    into    equal    parts    at     C,     Z>,     E,     and     F   (P.  XV., 

a  2). 


PRpBLE3I      II. 

To  construct  a  fourth  proportional  to  three  give7i  straight  lities. 

Let  Ay  By  and  (7,  be 
the  given  lines.  Draw 
BE  and  BF,  making 
any  convenient  angle  with 
each  other.  Lay  off  BA 
equal  to  A,  BB  equal 
to    B,     and     BC    equal 

to     C  ;     draw    AC,     and     from      B      draw    B^  parallel  to 
AC  :     then   will    BX^    be   the  fourth  proportional  required. 

For    (P.  XV.,  C),   we   have, 


i 


or. 


BA    :     BB    :  :    BC    :    BX ; 
A    :       B     :  :        C    :    BX. 


Cor.    If    BC    la  made    equal    to    BBy     BX    will    be 
third   proportional   to    BA     and    BB,     or    to    A     and    B. 


BOOK     IV. 


131 


PROBLEM     ni. 

To  constnid  a  mean  pi^oport L07ial  hetivcen   iioo  given  straigTit 

lines. 

Let  A  and  B  be  the  given 
lines.  On  an  indefinite  line,  lay  off 
DE  equal  to  A^  and  EF  equal 
to  ^ ;  on  DE  as  a  diameter  de- 
scribe the  semi-circle  DGE^  and 
draw  EG  perpendicular  to  DF : 
then   will    EG    be   the   mean  proportional  required. 

For   (P.  XXm.,  C.  2),  we  have, 


or, 


BE    \    EG    :  :    EG    :    EF  \ 
A    ',    EG    \\    EG    :    B, 


PROBLEM     IV. 


To  divide  a  given  straight  line  into  tiuo  such  parts,  that  the 
greater  part  shall  he  a  mean  proportional  letioeen  the  whole 
line  and  the  other  part. 

Let    AB    be  the  given  line. 

At  the  extremity  J?,  draw 
BO  perpendicular  to  AB,  and 
make  it  equal  to  half  of  AB. 
With  (7  as  a  centre,  and  CB 
aa  a  radius,  describe  the  aro 
JDBE  ;    draw    A  C,    and  produce 

it  tiU  it  terminates  in  the  concave  arc  at  E  ;  with  A  as 
centre  and  AB  as  radius,  describe  the  arc  BF :  then 
will    AF   be  the   greater  part   required. 


132 


GEOMETRY. 


For,    AB     bebig  perpendicular  to     CB     at     B,    is  tan. 
gent  to    the    arc    DBJEJ  :    hence 
(P.  XXX.), 

a 

AJi:    :    AB     I  :    AB     :    AB ; 
and,   by  division    (B.  IE.,   P.  YI.)>       ^ 


AE-  AB 


AB    :  :    AB  -  AD    :    AD. 


But,  BE  is  equal  to  twice  GB^  or  to  AB  :  hence, 
yl^  -  AB  is  equal  to  AD,  or  to  ^i^;  and  AB  —  AD 
is   equal  to    AB  —  AF,    or  to    FB  :     hence,   by  substitution. 


AF   :    AB 


FB    :    AF ; 


and,   by  inversion    (B.  11.,  P.  V.), 

AB    :    AF   :  :    AF   :    FB. 


Scholium.  When  a  straight  line  is  divided  so  that  the 
greater  segment  is  a  mean  proportional  between  the  whole 
line  and  the  less  segment,  it  is  said  to  be  divided  in  extreme 
and  711  can  ratio. 

Since  AB  and  DE  are  equal,  the  line  AE  is  divided  in 
extreme  and  mean  ratio  at  D ;  for  we  have,  from  the  first 
of  the  above  proportions,   by  substitution, 

AE  '.  DE  '.'.  DE  :    AD, 


BOOK    IV. 


133 


PROBLEM   V. 

Tliroiigh  a  given  point,  in  a  given  angle,  to  draw  a  straight 
line  so  that  the  segments  letween  tie  point  and  the  sides  of 
the  angle  shall  be  eq^ial.  • 

Let    BCD   be  the  given  angle,  and    A    the  given  point. 

Tlirough  A,  draw  A£J  parallel  to 
DC  ;  lay  off  BF  equal  to  CU,  and 
draw  FAD :  then  will  AF  and  AD 
be  the   segments   reqiiii'cd. 

For   (P.  XV.),   we  have, 


FA 


AD 


FE    :    EC', 


but,    FE    is  equal  to    EC  ;    hence,    FA    is  equal  to    AD. 


PROBLEM     VI. 
To  construct  a   triangle  equal  to  a  given  polygon. 

Let    ABCDE    be   the  given  polygon. 

Draw  CA  ;  produce  EA^  and 
draw  BG  parallel  to  CA  ;  draw 
the  line  CG.  Then  the  triangles 
BAC  and  GAC  have  the  com- 
mon base  ACy  and  because  their 
vertices      B      and      G      lie     in     the 

same  line  BG  parallel  to  the  base,  their  altitudes  are  equal, 
and  consequently,  the  triangles  are  equal  :  hence,  the  polygon 
GCDE    is   equal  to   the   polygon     ABCDE. 

Again,  draw  CE ;  produce  AE  and  draw  DF  parallel 
to  CE ;  draw  also  CF ;  then  will  the  triangles  FCE 
and  DCE  be  equal:  hence,  the  triangle  GCF  is  equal 
to  the  polygon  GCDE,  and  consequently,  to  the  given 
polygon.      In    like    manner,    a    triangle    may    be    con?tructed 


134 


GEOMETRY. 


PEOBLEM     Vn. 


To  construct  a  square  equal  to  a  giveyi  triangle. 

Let    ABC   be  the  given  triangle,     AD    its  altitude,   and 
jBC    its  base. 

Construct  a  mean  pro- 
portional between  AD 
and  Lalf  of  DC  (Prob. 
m.).  Let  XT  be  that 
mean  proportional,  and  on 
it,   as  a  side,   construct   a 

square :     then  will  this    be    the    square    required.      For,   from 
the   construction, 


F^t72 


Xy  =   ^BC  X  AD  =  area  ABC. 

Scholium.    By  means  of  Problems  VI.  and  Vll.,   a  square 
may    be   constructed   equal  to   any   given  polygon. 


PROBLEM     Vni. 


On  a  given  straigU  line,  to  construct  a  polygon  similar  to  a 

give7i  polygon. 

Let    FG    be    the    given    bjie,    and    ABODE   the    given 
polygon.      Draw    AC    and    AD. 

At  F^  construct 
the  angle  GFJI  equal 
to  BACy  and  at  G 
the  angle  FGH  equal 
to  ABC  ;  then  will 
FGU  be  similar  to 
ABC    (P   XVHL,  C.) 


BOOK    IV. 


135 


In  like  manner,  construct  the  triangle  Fill  similar  to  ACD, 
and  FIK  similar  to  ADE]  then  will  the  polygon  FGHIK 
be  similar  to  the  polygon  ABODE  (P.  XXVI.,   C.  3). 


PROBLEM     IX. 


To  construct  a  square  equal  to  the  sum  of  two  given 
squares :  also  a  square  equal  to  the  difference  of  two 
given  squares. 

1°.     Let    A     and    JB    be   the   sides    of  the   given   squares, 
and  let    A     be   the   greater. 

Construct  a  right  angle 
CDE  ;  make  EE  equal 
to  yl,  and  EC  equal  to 
B ;  draw  CE^  and  on  it 
construct  a  square :  this  square  wall  be  equal  to  the  sum 
of  the  given   squares    (P.  XI.). 

2®.     Construct   a  right   angle     CEE. 

Lay  oK  EG  equal  to  E  ;  with  C 
as  a  centre,  and  CE^  equal  to  A,  as 
a  radius,  describe  an  arc  cutting  EE  at 
E ;  draw  CE^  and  on  EE  construct 
a  square  :  this  square  will  be  equal  to 
the   difference   of  the   given   squares    (P.  XL,  C.  1). 

Scholhim.    A  polygon  may  be  constructed  similar  to  either 
'of  two  given  polygons,  and  equal  to  their  sum  or  difference. 

For,  let  A  and  B  be  homologous  sides  of  the  given  polygons 
Find  a  square  equal  to  the  sum  or  difference  of  the  squares 
on  A  and  E;  and  let  X  be  a  side  of  that  square.  On  X  as 
a  side,  homologous  to  A  or  B,  construct  a  polygon  similar 
to  the  given  polygons,  and  it  will  be  equal  to  their  sum  or 
difference   (F.  XXVIL,   C.   2). 


BOOK     y. 

BKQULAE      POLYGONS.  —  AREA      OF      THE      CIRCLE, 

DEFINITIOlSr. 

1.    A    Regular     Polygon    is    a    polygon    which    is    T)0th 
equilateral   and   equiangular. 


PEOPOSITION      I.        THEOKEJr. 
Regular  x>olygons  of  the  same  number  of  skies   are   similar. 

Let    ABCDEF   and    abcdef    be  regular  polygons  of  the 
same   number   of  sides  :    then    will   they   be    similar. 

For,  the  corresponding 
angles  in  each  are  equal, 
because  any  angle  in 
either  polygon  is  equal 
to  twice  as  many  right 
angles  as  the  polygon 
has   sides,   less  four  right 

angles,  divided  by  the  number  of  angles  (B.  I,  P.  XXYI , 
C.  4) ;  and  further,  the  corresponding  sides  are  proportional, 
because  all  the  sides  of  either  polygon  are  equal  (D.  1) :  hence, 
the  polygons  are  similar  (B.  IV.,  D.  1);  which  was  to  be  proved. 


BOOK     V. 


137 


1*R0P0SITI0IT      II. 


TITEOEEM. 


The  circumference  of  a  circle  may  he  circumscribed  about  any 
regular  polygon  /   a  circle  may  also  be  inscribed  in  it. 

l'      Let    ABCF    be    a    regular    polygon:    then    can    the 
circumference   of  a   circle   be   circumscribed   about   it. 

For,  through  three  consecutive  ver- 
tices A^  2?,  (7,  describe  the  circum- 
ference of  a  circle  (B.  III.,  Problem 
Xin.,  S.).  Its  centre  0  will  lie 
on  jPO,  drawn  perpendicular  to  JBC^ 
at  its  middle  point  P;  draw  OA 
and     OD. 

Let  the  quadrilateral  OPCD  be 
turned  about   the  line    OP,    until   PC 

falls  on  PB  ;  then,  because  the  angle  C  is  equal  to  P, 
the  side  CD  will  take  the  direction  BA  ;  and  because  CD 
is  equal  to  P^4,  the  vertex  P,  w^ill  faU  upon  the  vertex 
A  ;  and  consequently,  the  line  OD  will  coincide  with  OA.^ 
and  is,  therefore,  equal  to  it :  hence,  the  circumference  which 
passes  through  yl,  P,  and  C,  wiU  pass  through  D.  In 
like  manner,  it  may  be  shown  that  it  will  pass  through  all 
of  the  other  vertices :  hence,  it  is  circumscribed  about  the 
polygon  ;    which  was   to  he  proved. 


1".     A   circle   may  be   inscribed   in   the   polygon. 

P  ji\  the  sides  AB^  BC,  &c.,  being  equal  chords  o 
tne  ciicumscribed  circle,  are  equidistant  from  the  centre  0 
hence,  if  a  circle  be  described  from  0  as  a  centre,  with 
OP  as  a  radius,  it  will  be  tangent  to  all  of  the  sides  ci 
the  polygon,  and  consequently,  will  be  inscribed  in  it;  which 
was  to  be  proved. 


138 


GEOMETRY. 


Scholium.  If  the  circnmference  of  a  circle  be  divided 
into  equal  ai-cs,  the  chords  of  these  arcs  will  be  sides  of  a 
regular   inscribed   polygon. 

For,  the  sides  are  equal,  because  they  are  chords  of  equal 
arcs,  and  the  angles  are  equal,  because  they  'are  measured  by 
halves   of  equal   arcs. 

If  the  vertices  A^  D,  C,  &c., 
of  a  regular  inscribed  polygon  be 
joined  Avith  the  centre  0,  the  tri- 
angles thus  formed  will  be  equal, 
because  their  sides  are  equal,  each 
to  each  :  hence,  all  of  the  angles 
about  the  j^oint  0  are  equal  to 
vjach  other. 


DEFINTTIONS. 


1.    The   Centre   of   a  Regular  Polygon,   is  the   common 
centre   of  the   circumsci'ibed   and   inscribed   circles. 


2.  The  Angle  at  tile  Centre,  is  the  angle  formed  by 
dra^ving  lines  from  the  centre  to  the  extremities  of  either 
side. 

The  angle  at  the  centre  is  equal  to  four  right  angles 
divided   by   the   number   of  sides   of  the   polygon, 

3.  The  Apothem,  is  the  shortest  distance  from  the  centre 
to   cither   side. 

The  apothegm  is  equal  to  the  radius  of  the  inscribed 
circle. 


BOOK     V. 


139 


Pfl(,>r()SITION     in.        TROBLEM. 


To  inscribe  a  square  in  a  given  circle. 

Let  ABCD  be  tlie  given  cir- 
cle. Draw  any  two  diameters  AC 
and  JiD  perpendicular  to  each 
other  ;  tl/ey  will  divide  the  circum- 
ference into  four  equal  arcs  (B.  III., 
P.  XVIL,  S.).  Draw  the  chorda 
AB,  BG,  CB,  and  BA  :  then 
will  tlie  figure  ABCB  be  the 
square   required   (P.  II.,  S.). 

ScJiolium.    The    radius    is    to    the    side    of   the    inscribed 
square  as     1     is  to   -y^ 


PROPOSITION     rV.        THEOREM. 

If  a  regular  hexagon  he  inscribed  i}i   a   circle,  any  side  will 
be  equal  to   the  radiits   of  the  circle. 

Let  ABB  be  a  circle,  and  ABCBEIT  a  regular  in- 
scribed hexagon :  then  will  any  side,  as  AB,  be  equal  to 
the   radius  of  the   circle. 

Draw  the  radii  OA  and  OB. 
Then  will  the  angle  AOB  be 
equal  to  one-sixth  of  four  right 
angles,  or  to  two-thirds  of  one 
right  angle,  because  it  is  an  an- 
gle at  the  centre  (P.  II.,  D.  2). 
The  sum  of  the  two  angles  OAB 
and    OBA   is,  consequently,  equal 

to  four-thirds  of  a  right  angle  (B.  L,  P.  XXV.,  C.  1)  ;  but, 
the  angles  OAB  and  OBA  are  equal,  because  the  opposite 
sides     OB     and     OA    are    equal  :     hence,    each    is    equal    to 


140 


GEOMETRY. 


two-thirds  of  a  right  angle.  The   three  angles  of  the  triangle 

AOB    are    therefore,  equal,  and   consequently,   the   triangle  is 

equilateral :    hence,    AH    is  equal   to     OA  ;    which  was  to  he 
proved. 

PKOPOSITION     V.        PROBLEM. 
To  inscribe  a  regular  hexagon  in   a  given  circle. 

Let    ABE    be   a   circle,   and     0    its  centre. 

Beginning  at  any  point  of 
the  circumference,  as  A^  ap- 
ply the  radius  OA  six  times 
as  a  cliord  ;  then  will 
ABCDEF  be  the  hexagon 
required    (P.  IV.). 

Cor.  1.  If  the  alternate 
vertices  of  the  regular  hexagon 
be  joined  by  the  straight  lines 
A  C,  CE,  and  EA,  the  inscribed 
triangle  ACE  will   be   equilateral    (P.  II.,  S.). 

Cor.  2.  If  we  draw  the  radii  OA  and  OC,  the  figure 
AOCB  will  be  a  rhombus,  because  its  sides  are  equal: 
hence   (B.  IV.,  P.  XIV.,  C),  we  have, 

Jb"  +  BG^  +  OA^  -{-  OC^  =  AC'  +   OB' ; 

or,  taking   away  from  the  first    member    the    quantity      OA , 
and  from  the  second  its  equal     OB^,     and  reducing,  we  have 


dOA'  =   Ad 


whence   (B.  H.,  P  H.), 


AC' 


OA' 


1; 


BOOK     V. 


141 


or   (B.  n.,  P.  Xn.,  C.  2), 

AC    :     OA 


:     V^    :     1 ; 


Uiat  is,  the  side  of  an  inscribed  equilateral  triangle  is  to  the 
radius^  as   the  square  root  of  3   is  to   1. 


PEOPOSITION     VI.        THEOREM. 

J^  the  radius  of  a  circle  be  divided  in  extreme  and  mean 
ratio,  the  greater  segment  icill  be  equal  to  one  side  of  a 
regular  inscribed  decagon. 

Let  ACG  be  a  circle,  OA  its  radius,  and  AB,  equal  to 
OMj  the  greater  segment  of  OA  when  divided  in  extreme 
and  mean  ratio  :  then  will  AB  be  equal  to  the  side  of  a 
regular   inscribed   decagon. 

Draw   OB   and  BM.    We 
hare,  by  hy|)othesi3, 


AO  :    OM 


OM  :   AM: 


or,    since     AB     is    equal    to 
OM,    we  have, 


AO   :   AB 


AB   :  AM\ 


hence,  the  triangles  OAB 
and  BAM  have  the  sides 
about    their    common     angle 

BAM,  proportional  ;  they  are,  therefore,  similar  (B.  IV., 
P.  XX.).  But,  the  triangle  OAB  is  isosceles  ;  hence,  BAM 
is  also  isosceles,  and  consequently,  the  side  BM  is  equal  to 
AB.  But,  AB  is  equal  to  031,  by  hypothesis  :  hence, 
BM    is   equal  to     031,     and   consequently,   the   angles    MOB 


142 


GEOMETRY. 


and  MBO  are  equal.  The  angle  AMB  being  an  exterior 
angle  of  the  triangle  OMB^  is  equal  to  the  sum  of  the 
angles  MOB  and  MBO^  or 
to  twice  the  angle  MOB  ; 
and  because  AMB  is  equal  to 
OAB,  and  also  to  OBA,  the 
eura  of  the  angles  OAB  and 
OBA  is  equal  to  four  times 
the  angle  A  OB  :  hence,  A  OB 
is  equal  to  one-fifth  of  two 
right  angles,  or  to  one-tenth  of 
four  right  angles  ;  and  -  conse- 
quently, the  arc  AB  is  equal 
to  one-tenth  of  the  circumfer- 
ence :  hence,  the  chord  AB  is  equal  to  the  side  of  a 
regular  inscribed  decagon  ;    which  was   to   be  proved. 

Cor.  1.  If  AB  be  applied  ten  times  as  a  chord,  the 
resulting  polygon   will   be   a   regular   inscribed   decagon. 

Cor.  2.  If  the  vertices  A,  (7,  ^,  G,  and  2',  of  the 
alternate  angles  of  the  decagon  be  joined  by  straight  lines, 
the   resulting  figure   will  be  a  regular  inscribed   pentagon. 


Scholium  1.  If  the  arcs  subtended  by  the  sides  of  any 
regular  inscribed  polygon  be  bisected,  and  chords  of  the  semi- 
arcs  be  drawn,  the  resulting  figure  will  be  a  regular  inscnooc' 
polygon  of  double   the  number   of  sides. 

Scholium  2.  The  area  of  any  regular  inscribed  polygon 
is  less  than  that  of  a  regular  inscribed  polygon  of  double 
the  number  of  sides,   because  a  part  is  less  than   the   whole 


BOOK    V. 


143 


PROrOSITION"   VII.      TROBLEM. 

To    circtimscrile,    almit    a    circle,    a    polygon    which    shall    be 
similar   to  a  given  regular  inscribed  iwlygon. 

Let  TKQ  be  a  circle,  0  its  centre,  and  ABCDEF 
a  regular  inscribed  polygon. 

At  the  middle  points 
r,  iV,  P,  &c.,  of  the  arcs 
subtended  by  the  sides  of 
the  inscribed  polygon,  draw 
tangents  to  the  circle,  and 
prolong  them  till  they  in- 
tersect ;  then  will  the  re- 
sulting figure  be  the  poly- 
gon required. 

1°.  The  side  HQ  be- 
ing   parallel    to    BA^     and 

HI  to  i>C,  the  angle  H  is  equal  to  the  angle  B.  In 
like  manner,  it  may  be  shown  that  any  other  angle  of  the 
circumscribed  polygon  is  equal  to  the  corresponding  angle  of 
the  inscribed  polygon :  hence,  the  circumscribed  polygon  iii 
equiangular. 

2°.  Draw  the  straight  lines  OG,  OT,  OR,  ON,  and  01. 
Then,  because  the  lines  HT  and  HN  -are  tangent  to  the 
circle,  OH  will  bisect  the  angle  NHT,  and  also  the  angle 
NOT  (B.  III.,  Prob.  XIV.,. S.);  consequently,  it  will  pass 
through  the  middle  point  B  of  the  arc  NBT.  In  like 
manner,  it  may  be  shown  that  the  straight  line  drawn 
fiom  the  centre  to  the  vertex  of  any  other  angle  of  the 
circumscribed  polygon,  will  pass  through  the  corresponding 
vertex  of  the  inscribed  polygon. 

The    triangles    OHG    and     OHI    have    the    angles    OHO 


U4 


GEOMETPwY. 


and  OUT  eijual,  from  what  lias  just  been  shown  ;  the  an- 
gles GOn  -AvA  IIOI  equal,  because  they  are  measured  by 
the  equal  arcs  AB  and 
BC,  and  the  side  OH 
winnion  ;  they  are,  there- 
fore, equal  in  all  their 
parts  :  hence,  GH  is 
equal  to  III,  In  like 
manner,  it  may  be  sho^^^l 
that  III  is  equal  to  IK^ 
IK  to  JTZ,  and  so  on  : 
hence,  the  circumscribed 
polygon   is   equilateral. 

The  circumscribed  poly- 
gon being   both   equiangular   and   equilateral,   is  regular  /    and 
since  it   has  the   same   number  of  sides   as   the  inscribed  poly- 
gon, it   is   similar   to   it. 

Cor.  1.  If  straight  lines  be  drawn  from  the  centre  of  a 
regular  circumscribed  polygon  to  its  vertices,  and  the  consec- 
utive points  in  which  they  intersect  the  circumference  be 
joined  by  chords,  the  resulting  figure  will  be  a  regular 
inscribed  polygon   similar   to   the   given   polygon. 

Cor.  2.  The  sum  of  the  lines  HT  and  iZZV  is  equal 
to  the  sum  of  HT  and  TG^  or  to  HG  ;  that  is,  to  one 
of  the   sides   of  the   circumscribed   polygon. 

Cor.  3.  If  at  the  vertices  -4,  B^  (7,  &c.,  of  the  in- 
scribed polygon,  tangents  be  dra"\vn  to  the  circle  and  pro- 
longed till  they  meet  the  sides  of  the  circumscribed  polygon, 
the  resulting  figure  ■will  be  a  circumscribed  polygon  of  double 
the   number  of  sides. 


Cor.  4.    The    area    of  any  regular  circumscribed    polygon 


BOOK     V. 


145 


is  greater  tlian  tliat  of  a  regular  circumscribed  polygon  of 
double  the  number  of  sides,  because  the  whole  is  greater 
than   any   of  its  parts. 

ISchoUum.  By  means  of  a  circumscribed  and  inscribed 
square,  we  may  construct,  in  succession,  regular  circumscribed 
and  inscribed  polygons  of  8,  16,  32,  &c.,  sides.  By  means 
of  the  regular  hexagon,  we  may,  in  like  manner,  construct 
regular  polygons  of  12,  24,  48,  &c.,  sides.  By  means  of  the 
decagon,  we  may  construct  regular  polygons  of  20,  40,  80, 
&c.,   sides. 


PROPOSITIOlSr     VIII.        THEOEEM. 


The  area  of  a  regular  ijolygon  is  equal  to  half  the  product 
of  its  perimeter  and  apothem. 

Let  GLTIK  be  a  regular  polygon,  0  its  centre,  and 
OT  its  apothem,  or  the  radius  of  the  inscribed  circle : 
then  will  the  area  of  the  polygon  be  equal  to  half  the 
product   of  the   perimeter  and   the   apothem. 

For,  draw  lines  from  the  centre 
to  the  vertices  of  the  polygon. 
These  lines  will  divide  the  polygon 
into  trianffles  whose  bases  will  be 
the  sides  of  the  polygon,  and 
whose  altitudes  will  be  equal  to 
the  apothem.  Now,  the  area  of 
any  triangle,  as  OIIG^  is  equal  to 
half  the  product  of  the  side  HG 
and  the   apothem :    hence,    the    area 

of  the  polygon  is  equal   to  half  the   product  of  the  perimeter 
and   the   apothem ;    xohich  ica^   to   he  proved. 

10 


146 


GEOMETRY. 


PKOPOSITION      IX.         THEOREM. 

T?ie  perimeters  of  similar  regular  polygons  are  to  each 
other  as  the  radii  of  their  circumscribed  or  inscribed 
circles  ;  and  their  areas  are  to  each  other  as  the  squares 
of  those  radii. 

1°.  Let  ABC  and  KLM  be  similar  regular  polygons. 
Let  OA  and  QK  be  the  radii  of  their  circumscribed,  OD 
and  QR  be  the  radii  of  their  inscribed  circles :  then  will 
the  perimeters  of  the  polygons  be  to  each  other  as  OA  is 
.0     QK^    or  as     OD    is   to     QB. 

For,   the    lines 
i)A   and   QK  are  K^ — =n — -J5 

homologous  lines 
iii  the  polygons 
;'.o  which  they  be- 
long, as  are  also 
iJie  lines  OD  and 
QB  :  hence,  the 
()erimeter  of  AB  G 

is  to  the  perimeter  of  KLM,  as  OA  is  to  QK,  or  as 
OD  is  to  QB  (B.  IV.,  P.  XXVII.,  C.  1) ;  which  was  to  be 
\yroved. 

2°.  The  areas  of  the  polygons  will  be  to  each  other  as 
OA^    is  to     QK\    or   as    Olf    is  to    QB". 

For,  OA  being  homologous  with  QK,  and  OD  with 
QB,  we  have,  the  area  of  ABC  is  to  the  area  of  KLM 
as  OA"  is  to  QK^,  or  as  OD"  is  to  QB"-  (B.  IV.,  P 
XXV n.,  C.  1)  ;    which  was  to  be  proved. 


BOOK     V. 


Ul 


PROPOSITION     X.        THEOREM. 


7\co  regular  polygons  of  the  same  niimber  of  sido.s  can  bt 
constructed^  the  one  circumscribed  about  a  circle  and  the 
other  inscribed  in  it^  which  shall  differ  from  each  oihei 
by  less  than  any  given  surface. 

Let  ABCE  be  a  circle,  0  its  centre,  and  Q  the  side  of 
a  square  equal  to  or  less  than  the  given  surface;  then  can 
two  dmilar  regular  polygons  be  constructed,  the  one  circum^ 
scribed  about,  and  the  other  inscribed  within  the  given  tiircle, 
which  shall  differ  from  each  other  by  less  than  the  square 
ot    $,     and  consequently,   by  less  than  the  given   surface. 

Inscribe  a  square  in  the 
given  circle  (P.  HI.),  and  by 
means  of  it,  inscribe,  in  succe8- 
don,  regular  polygons  of  8,  16, 
32,  &c.,  sides  (P.  VIE.,  S.),  un- 
til one  is  found  whose  side  is 
less  than  Q ;  let  AB  be  the 
side  of  such  a  polygon. 

Construct  a  similar  circum- 
scribed polygon  abode  :  then 
will  these  polygons   differ  from   each    other  by  less    than    the 

square   of    Q. 

For,  from  a  and  5,  draw  the  lines  a  0  and  b  0 ;  they 
will  pass  through  the  points  A  and  B.  Draw  also  OK 
to  the  point  of  contact  K',  it  will  bisect  AB  at  I  and 
be  perpendicular  to  it.      Prolong    AO    to     E. 

Let  F  denote  the  circumscribed,  and  p  the  inscribed 
polygon ;  then,  because  they  are  regular  and  similar,  we 
stiaU  have   (P.  IX.), 


148 


GEOMETRY. 


P    '.    p    '.:     OK     or     OA     : 
hence,   by  divdsion   (B.  11.,  P.  VI.),   we  have, 


0-f'. 


P  -p    : 


or, 


P    :    P  -p    :  :     OA' 


OA"" 


AI\ 


OA'  -  or 


Maltiplying  the  terms  of  the 
second  couplet  by  4  (B.  11.,  P. 
VII),   we  have, 

P   :    P-p    :  :    4  04"    :    4.IT' ; 

whence   (B.  IV.,  P.  VHI.,  C), 


P    :     P-p    :  :     AB' 


AB\ 


But    P    is  less   than   the   square   of   A^E   (P.  VQ.,  C.  4)  ; 
hence,     P  —  p    is  less  than   the   square   of   AP,     and   conse 
quently,   less  than   the   square   of    Q,     or  than   the   given   sur- 
face ;    lohich  was   to  be  proved. 

Pefiiiition. — The  liinit  of  a  variable  quantity  is  a  quantity  to- 
wards which  it  may  be  made  to  approach  nearer  than  any  given 
quantity,  and  which  it  reaches  under  a  particular  supposition. 

Lemma. — Tico  variable  quantities  xohich  constantly  approach 
towards  equality,  and  of  lohich  the  difference  becomes  less  than 
any  finite  magnitude,  are  xdtimately  equal. 

For  if  they  are  not  ultimately  equal,  let  D  be  their  ultimate 
difference.  Now,  by  hypothesis,  the  quantities  have  approached 
nearer  to  equality  than  any  given  quantity,  as  D\  hence  D 
denotes  their  difference  and  a  quantity  greater  than  their  differ- 
ence, at  the  same  time,  whicli  is  impossible ;  therefore,  the  two 
quantities  are  ultimately  equal.* 

*  Newtou's  Priucipia,  Book  I.,  Lemma  I. 


BOOK    V. 


149 


Cor.  If  we  take  any  two  similar  regular  polygons,  the  one  cir- 
cumscribed about,  and  the  other  inscribed  within  the  circle,  and 
bisect  the  arcs,  and  then  circumscribe  and  inscribe  two  rerrular 
polygons  having  double  the  number  of  sides,  it  is  plain  that  by 
continuing  the  operation,  two  new  polygons  may  be  found  which 
shall  differ  from  each  other  by  less  than  any  given  surface  ; 
hence,  by  the  lemma,  the  two  polygons  will  become  ultimately 
equal.  But  this  equality  cannot  take  place  for  any  finite  number 
of  sides ;  hence,  the  number  of  sides  in  each  will  be  infinite,  and 
each  will  coincide  with  the  circle,  which  is  their  common  limit. 
Under  this  hypothesis,  the  i^erimeter  of  each  polygon  will  coin- 
cide with  the  circumference  of  the  circle. 

Scholium. — The  circle  may  be  regarded  as  a  regular  polygon 
having  an  infinite  number  of  sides.  The  circumference  may  be 
regarded  as  the  ^)er/??ie?er,  and  the  radius  as  the  apothem. 

PROPOSITION     XI.        PEOBLEM. 

The  area  of  a  regular  inscribed  polygon^  and  that  of  a 
similar  circumscribed  polygon  being  given.,  to  find  the 
areas  of  the  regular  inscribed  and  circumscribed  polygons 
having  double  the  number  of  sides. 

Let  AJ^  be  the  side  of  the  given  inscribed,  and  EF 
that  of  the  given  circumscribed  polygon.  Let  C  be  their 
common  centre,  A3IB  a  portion  of  the  circumference  of 
the   circle,   and    M    the   middle   point   of  the   arc    A  MB. 

Draw  the  chord  A3f,  and 
at  A  and  J5  draw  the  tangents 
AP  and  BQ\  then  will  AM 
be  the  side  of  the  inscribed 
polygon,  and  PQ  the  side  of 
the  circumscribed  polygon  of 
double  the  number  of  sides  (P. 
Vn.).  Draw  CE,  CP,  CM, 
and     CF. 


150 


GEOMETRY. 


Denote  the  area  of  the  given  inscribed  polygon  by  jt>, 
the  area  of  the  given  circumscribed  polygon  by  P,  and  the 
areas  of  the  inscribed  and  circumscribed  polygons  having 
double   the   number   of  sides,   respectively   by   p'     and    P' . 

V.  The  triangles  GAB,  CAM, 
and  CB3f,  are  like  parts  of ^  the 
polygons  to  which  they  belong  : 
hence,  they  are  proportional  to  the 
polygons  themselves.  But  CAM 
is  a  mean  proportional  between 
CAB  and  CU3I  (B.  lY.,  P. 
XXrV.,  C)  ;  consequently  p' 
is  a  mean  proportional  between 
p    and    P:    hence. 


p'  =  ^p  X  p. 


(1.) 


2°.  Because  the  triangles  CPM  and  CPE  have  the 
common  altitude  (7J/,  they  are  to  each  other  as  their 
bases  :    hence, 

CPM    :     CPE    :  :    PM    :    PE  \ 
and  because    CP    bisects  the  angle   ACM,    we  have  (B.  IV., 

P.  xvn.), 


PM   :    PE    :  :     CM    :     CE 
hence   (B.  II.,   P.  IV.), 

CPM    :     CPE    :  :     CB     : 


CB 


CA; 


CA     or     CM. 


But^    the    triangles  CAB     and    CAM     have    the    common 

altitude    AB  ;     they  are     therefore,    to    each    other    as    theii 
bases  :    hence, 

CAB     :  CAM    :  :     CB     :     CM-, 


BOOK     V.  161 

or,    because     CAD    and    CAM     are    to    each    other    as    the 
polygons  to   which   they  belong, 

p     :     p'     :  :     CD     :     CM  ] 
hence    (B.  II.,   P.  IV.),   we   have, 

CF3I    :     CPE    \  '.     p     \     v\ 
and,   by  composition, 

CPM   :    CP3I+  CPE    or    CME     :  :    p    :    p  ^  p'  ; 
hence    (B.  II.,   P.  VH.), 

2  CPM    or    CMPA    :    CME    :  :    2p     :     p  + ;/. 

But,     CMPA      and     C3fB     are    like    parts    of    P'     and     P, 

hence, 

P'    :    P    :  :    2p    :    p  -{-  p'  \ 


or, 


r  =  ^^4^ (2.) 

p  +  p' 

Scholium.    By  means  of  Equation    ( 1 ),     we   can    find    p\ 
and   then,   by  means   of   Equation   ( 2 ),   we  can   find    P'. 


PROPOSITION     XII.        PIIOBLEM. 

To  find  the  approximate  area   of  a  circle  lohose  radius  is  1. 

The  area  of  an  inscribed  square  is  equal  to  twice  the  square 
described  on  the  radius  (P.  III.,  S.),  wliich  square  ii  ilie  iniii 
of  measure,  and  is  denoted  by  1.  The  area  of  the  circumscribed 
square  is  4.  Making  p  equal  to  2,  and  P  equal  to  4,  we  have, 
from  Equations  (1)  and  (2)  of  Proposition  XL, 


p'     =     VS        =  2.8284271    .    .    .    inscribed  octagon; 

>/    

2  4-  \/S 


1  c 

P'    — —  =  3.3137085    .    .    .    circumscribed  octagon. 


152 


GEOMETRY. 


Making    p     equal   to   2.8284271,    aud    P    equal   to   3.3137085, 
we   Lave,   from  the   same   equations, 

p'  =    3.0614674     .     ,     .     inscribed  polygon  of  16  sides. 

P'  =    3.1825979     .     .     .     circumscribed  polygon  of  16  sidea 

By   a    continued    application    of    these    equations,    we     find 
the   areas  indicated   below. 


Ni'MBER   OF   Srosa. 

Inscribed  Polvgo.\3. 

Circumscribed  Polygons. 

4 

2.0000000 

4.0000000 

8 

2.8284271 

3.3137085 

16 

3.0614674 

3.1825979 

82 

3.1214451 

3.1517249 

64 

3.1365485 

3.1441184 

128 

3.1403311 

3.1422236 

256 

3.1412772 

3.1417504 

512 

3.1415138 

3.1416321 

1024 

3.1415729 

3.141G025 

2048 

3.1415877 

3.1415951 

4096 

3.1415914 

3.1415933 

8192 

3.1415923 

3.1415928 

16384 

3.1415925 

3.1415927 

Now,  the  figures  Ayhich  express  the  areas  of  the  two  last 
polygons  are  the  same  for  six  decimal  places;  hence,  those  areas 
differ  from  each  other  by  less  than  one-millionth  of  the  measuring 
unit.  But  the  circle  differs  from  either  of  the  jwlygons  by  les£ 
than  they  differ  from  eacli  other.  Hence,  1=  taken  3.141592  times, 
expresses  the  area  of  a  circle  whose  radius  is  1,  to  less  than  one« 
millionth  of  the  measuring  unit ;  and  by  increasing  the  number 
of  sides  of  the  polygons,  we  should  obtain  an  area  still  nearer  tlie 
true  one.  Denote  the  number  of  times  Avhicli  the  square  of  tlie 
radius  is  taken.,  by  *,  we  have, 

*  X  r  =  3.141592; 
that  id,  the  area   of  a  circle  whose  radius  is  1,  is  3.141592,  in 
%ohicU  the  unit  of  measure  is  tlie  square  on  the  radius. 

S:h.    For  ordinary  accuracy,  *  is  taken   equal  to  3.1 41G. 


BOOK     V. 


153 


PEOPOSITION      XIII.      THEOREM. 

The  circumferences  of  circles  are  to  each  other  as  their  radii^ 
and  the  areas  are  to  each  other  as  the  squares  of  their 
radii. 

Let  C  and  0  be  the  centres  of  two  circles  avIioho 
radii  are  CA  and  01)  :  then  will  the  circumferences  bo 
to  each  other  as  their  radii,  and  the  areas  will  be  to  each 
other  as  the   squares   of  their   radii. 


For,  let  similar  regular  polygons  MKPST  and  EFGKL 
be  inscribed  in  the  circles  :  then  will  the  perimeters  of  these 
polygons  be  to  each  other  as  their  aj)othenis,  and  the  areas 
will  be  to  each  other  as  the  squares  of  their  apothems,  what- 
ever may  be   the   number   of  their   sides    (P,  IX.). 

If  the  number  of  sides  be  made  infinite  (P.  X.  Scli.),  the 
polygons  will  coincide  with  the  circles,  the  perimeters  with 
the  circumferences,  and  the  apothems  Avith  the  radii  :  hence, 
ihe  circumferences  of  the  circles  are  to  each  other  as  their 
radii,  and  the  areas  are  to  each  other  as  the  squares  of  the 
radu  ;    xohich  was   to   he  proved. 

Cor.  1.  Diameters  of  circles  are  proportional  to  their 
radii :  hence,  the  circumferences  of  circles  are  proportional 
to  their  diameters^  and  the  areas  are  proportional  to  the 
squares   of  tht   diatneters. 


154 


GEOMETRY. 


Cor,  2,  Similar  arcs,  as  AB  and  DE^  are  like  parts 
of  the  circumferences  to  which 
they  belong,  and  similar  sectors, 
as  A  CB  and  D  OEy  are  like 
parts  of  the  circles  to  which 
they  belong  :  hence,  similar 
arcs  are  to  each  other  as  their 
radii^  and  similar  sectors  are 
to  each  other  as  the  squares  of  their  radii. 

Scholium.  The  term  infinite^  used  in  the  proposition,  js  to 
be  understood  in  its  technical  sense.  When  it  is  proposed  to 
make  the  number  of  sides  of  the  polygons  infinite,  by  the 
method  indicated  in  the  scholium  of  Proposition  X.,  it  is  sim- 
ply meant  to  express  the  condition  of  things,  when  the  in- 
scribed polygons  reach  their  limits ;  in  which  case,  the  dif- 
ference between  the  area  of  either  circle  and  its  inscribed 
polygon,  is  less  than  any  appreciable  quantity.  We  have  seen 
(P.  XII.),  that  when  the  number  of  sides  is  16384,  the  areas  differ 
by  less  than  the  millionth  part  of  the  measuring  unit.  By  increas- 
ing the  number  of  sides,  we  approximate  still  nearer. 


PROPOSITION     XIV. 


THEOEEM. 


The    area    of   a    circle  is  equal  to  half  the  product  of  its 
circumference  and  radius. 


Let     0    be    the   centre    of   a    circle, 
A  GBJE     its    circumference  :    then  will 
the  area  of  the  circle  be   equal   to  half 
the  product    of    the    circumference    and 
radius. 

For,  inscribe  in  it  a  regular  poly- 
gon ACDE.  Then  will  the  area  of 
thifi  polygon  be   equal   to  half  the  pro- 


OG    its   radius,    and 


BOOK     V.  155 

duct    of   its    perimeter    and    apothem,    whatever    may  be   the 
number  of  its   sides   (P.  "VTU.). 

K  the   number   of  sides   be   made  infinite,  the  polygon  will 

coincide  with  the  circle,  the   perimeter  with  the  circumlcreuco, 

and   the   apothem   with   the    radius  :     hence,   the    area   of   tht 

nrcle   is   equal  to   half  the   product   of   its    circumference   anl 

adius  ;    which  was   to  be  proved. 

Cor.  1.  The  area  of  a  sector  is  equal  to  half  the  pro- 
duct  of  its   are   and  radius. 

Cor.  2.  The  area  of  a  sector  is  to  the  area  of  the  circle, 
as  the   arc   of  the   sector   to   the   circumference. 

PROPOSITIOISr     XV.         PROBLEM. 

To  find  an  expression  for   the  area  of   any  circle    ifi    terms 

of  its  radius. 

Let     C    be   the   centre    of    a    circle,   and     CA     its    radius. 
Denote  its  area  by    area   CA,     its  radius 
by  Jt,     and  the   area   of   a   circle  whose 
radius  is   1,  by  -rr  x  1'  (P.  XII.,  S.). 

Then,  because  tlie  areas  of  circles 
are  to  each  other  as  tlie  squares  of  their 
radii   (P.  XIIL),  we  have, 

area  CA    :    -^r  x  1'    :  :    E'    :    1 ; 
whence,  area  CA    =   irH^. 

That  is,   the  area   of  any  circle  is   3.1416    times    the    square 
of  the  radiiis 

PROPOSITION     XVI.        PROBLEM. 

To  find  an  exjyression  for  the  circimference  of   a   circle^   m 
terms    of  its  radius,   or  diameter. 

Let     C     be  the   centre   of    a   circle,    and    CA     its   radius. 


156  GEOMii:TRY. 

Deuote  its  circumference  by   circ.  GA,    its  radius  by   i2,    and 
its  diameter  by    D.      From  the  last  Troposition,   we  have, 

area  CA    =    ifH^  ; 

and,    from   Proposition    XIV.,  we    have, 

area  CA   =   ^circ.  CA   x  B  \ 

hence,  ^circ.  CA  x  R   =  *i2*  ; 

whence,  by  redaction, 

circ.  CA   =   2*i2,    or,    circ.  CA   =   *i>. 

That    is,  the  circumference   of  any  circle  is   equal  to   3.1416 
times  its  diameter. 

Scholium  1.  The  abstract  number  -r,  equal  to  3.1416,  de- 
notes the  number  of  times  that  the  diameter  of  a  circle  is 
contamed  in  the  circumference,  and  also  the  number  of  tunes 
that  the  square  constructed  on  the  radius  is  contained  in  the 
area  of  the  circle  (P.  XV.).  Now,  it  has  been  proved  by 
the  methods  of  Higher  Mathematics,  that  the  value  of  -rr  is 
incommensurable  with  1  ;  hence,  it  is  impossible  to  express, 
by  means  of  numbers,  the  exact  length  of  a  circumference 
in  terms  of  the  radius,  or  the  exact  area  in  terms  of  tho 
square  described  on  the  radius.  We  may  also  infer  that  it 
:s  impossible  to  square  the  circle ;  that  is,  to  construct  a 
square  whose  area  shall  be  exactly  equal  to  that  of  the  circ.e. 

Scholium  2.  Besides  the  approximate  value  of  if,  3.1416, 
usually  employed,  the  fractious  ^  and  ffl  are  also  used  to 
express  the   ratio   of  the  diameter  to   the  circumference. 


BOOK     VI. 

PLANES        AND        POLYEDRAL        ANGLES, 

DEFI]SnTIOKS. 

1.  A  Straight  line  is  perpendicular  to  a  plane,  when 
it  is  perpendicular  to  every  straight  line  of  the  plane  which 
passes  through  its  foot;  that  is,  through  the  point  in  which 
it  meets  the  plane. 

In   this   case,   the   plane   is   also   perpendicular  to   the   line. 

2.  A  straight  line  is  parallel  to  a  plane,  when  it  can- 
not meet  the   plane,  how  for   soever  both  may  be  produced. 

In   this   case,   the   plane   is  also   parallel   to   the   line. 

3.  Two  Planes  are  parallel,  when  they  cannot  meet, 
how  far  soever   both   may  be   produced. 


4.  A  Diedral  angle  is  the  amount  of  divergence  of  two 
planes. 

The  line  in  which  the  planes  meet,  is  called  the  edge  of 
the  angle^   and  the  planes   themselves   are   called  faces   of  the 

angle. 

The  measure  of  a  diedral  angle  is  the  same  as  that  of  a 
plane  angle  formed  by  two  straight  lines,  one  in  each  face, 
and  both  perpendicular  to  the  edge  at  the  same  point.  A 
diedral  angle  may  be  ac^de,  oltuse,  or  a  rigid  angle.  In  the 
latter  case,  the  faces  are  perpe7iclicular  to   each   other. 


158 


GEOMETRY. 


5.     A  PoLYEDEAL  ANGLE  is  the   amount   of    divergence   of 
several  planes  meeting  at   a   common   point. 

This  point  is   called  the  vertex  of  tJie  angle  /    the  lines  in 
which    the    planes    meet    are    called   edges  of   the   angle^   and 
the    portions    of    the    planes    lying    between    the    edges    are 
called  faces  of  the  angle.    Thus,    S 
is  the  vertex  of  the  polyedral  angle, 
whose     edges    are    SA^      SB,     SC, 
SD,      and   whose    faces    are     AS-B, 
J5SC,     CSD,    DSA. 

A  polyedral  angle  which  has  but 
three  faces,  is  called  a  triedral 
angle. 


POSTULATE. 


A  straight  line  may  be  drawn  perpendicular  to  a  plane  from 
any  point  of  the  plane,  or  from  any  point  without  the  plane. 


PROPOSITION     I.        THEOREM. 

If  a  straight  line  has  two  of  its  points  in  a   plane,   it  will 

lie  wholly  in  that  plane. 

For,  by  definition,  a  plane  is  a  surface  such,  that  if  any 
two  of  its  points  be  joined  by  a  straight  line,  that  line  will 
lie   wholly   in   the   surface   (B.  L,  D.  8). 

Cor.  Through  any  point  of  a  plane,  an  mfinite  number 
of  straight  lines  may  be  drawn  which  will  lie  in  the  plane. 
For,  if  a  straigh-t  line  be  drawn  from  the  given  point  to  any 
other  point  of  the  plane,  that  line  will  lie  wholly  in  the  plane. 

Scholium.  If  any  two  points  of  a  plane  be  joined  by  a 
straight  line,  the  plane  may  be  turned  about  that  line  as  an 


BOOK    YI. 


159 


axis,  so  as  to  take  an  infinite  number  of  positions.  Hence, 
we  infer  that  an  infinite  number  of  planes  may  be  passed 
through  a  given  straight  line. 


PROPOSITION     n. 


THEOREM. 


Through    three   points^    not    in    the   same   straight    line,  one 
plane  can  be  passed,   and  only  one. 

Let  A,  J3,  and  C  be  the  three  points :  then  can  one 
plane   be   passed  through  them,   and  only  one. 

Join  two  of  the  points,  as  A  and 
B,  by  the  Une  AB.  Through  AB 
let  a  plane  be  passed,  and  let  this  plane 
be  turned  around  AB  until  it  contains 
the  point  C  ;  in  this  position  it  will 
pass  through  the  three  points  A,  B, 
and    C.       If   now,    the    plane    be   turned 

about  AB,  in  either  direction,  it  will  no  longer  contain  the 
point  C  :  hence,  one  plane  can  always  be  passed  through 
three  points,   and   only   one  ;    which  teas   to   be  proved. 

Cor.  1.  Three  points,  not  in  a  straight  line,  determine  the 
position  of  a  plane,  because  only  one  plane  can  be  passed 
through   them. 

Cor.  2.  A  straight  line  and  a  point  without  that  line, 
determine  the  position  of  a  plane,  because  only  one  plane 
can   be   passed   through  them. 

Cor.  3.  Two  straight  lines  which  intersect,  determine  the 
position  of  a  plane.  For,  let  AB  and  AC  intersect  at 
A  :  then  will  either  fine,  as  AB,  and  one  point  of  the 
other,   as    C,     determine  the  position   of  a  plane. 

Cor.  4.    Two  parallel  straight  lines  determine  the  position  of  a 


160 


GEOMETRY. 


plane.       For,    let    AB    and     CD    be   parallel.       By   definition 
(B.  I.,  D.  10)   two   parallel  lines  always  lie    in  the  same  plane. 
But   either   line,   as    AJ3,     and    any   point 
of   the   other,  as    I\     determine   the   posi- 
tion    of    a    plane  :     hence,    two    parallels 
determine   the   position   of  a  plane. 


A- 
C- 


F 


T 
L 


PROPOSITIOlSr      III. 


THEOEEM. 


The  intersection  of  two  planes  is  a  straight  line. 

Let    AB    and    CD    he  two   planes:    then  will  their  inter- 
section  be   a   straight  line. 

For,  let  E  and  F  be  any  two 
points  common  to  the  planes;  draw 
the  straight  line  EF.  This  line  hav- 
ing two  points  in  the  plane  AB, 
will  lie  wholly  in  that  plane  ;  and 
having   two   points   in   the    plane    CD, 

will  lie  wholly  in  that  plane  :  hence,  every  point  of  EF  is 
common  to  both  planes.  Furthermore,  the  planes  can  have 
no  common  point  lying  without  EF,  otherwise  there  would 
be  two  planes  passing  through  a  straight  line  and  a  point 
lying  without  it,  which  is  impossible  (P.  11.,  C.  2)  ;  hence, 
the  intersection  of  the  two  planes  is  a  straight  line  ;  which 
was  to  he  proved. 


PROPOSITION      IV.        THEOREM. 

If  a  straight  line  is  perpendicular  to  two  straight  lines  at 
their  point  of  intersection,  it  is  perpendicxdar  to  the  plane 
of  those  lines. 

Let    3IJSF    be  the  plane   of  the  tAvo  lines    BB,    CC,    and 
let    AP    be   perpendicular   to   these   lines    at    P  :      then    will 


BOOK    VI. 


161 


AP  be  perpendicular  to  every  straight  line  of  the  plane  which 
passes  through  F,  and  consequently,  to  the  plane  itself. 

For,  through  J*j  draw  in 
the  plane  MN,  any  line  PQ ; 
through  any  point  of  this  line, 
as  Q^  draw  the  line  J^C,  so 
that  JBQ  shall  be  equal  to  QC 
(B.  IV.,  Prob.  V.)  ;  draw  AJ3, 
A  Q,    and    A  C. 

The  base    J3C,     of  the   triangle    BPC,    being  bisected  at 
Q,    we  have    (B.  IV.,  P.  XIV.), 

PO"  +  PB"  =  'IPQ"  +  2QG\ 
In  like   manner,   we   have,   from  the   triangle    ABC, 


AC  +  AP'  =  2AQ'  +  2QC\ 

Subtracting    the    first    of    these    equations    from    the    second, 
member  from  member,   we  have. 


AC^  -  PC  +  AP""  -  PP"  =  ^AQ"  ^  2PQ\ 
But,   from  Proposition  XI.,  C.  1,  Book  IV.,   we  have, 

AC^  _  PC  =  AP\      and      AP"  -  PP^  =  AP  ; 
hence,   by   substitution, 

2PQ' ; 


whence, 


2AP'  =  2AQ' 


AW  =  AQ"  -  PQ'i 


or. 


AP'  +  PQ'  =  A  Q\ 


The  triangle  APQ  is,  therefore,  right-angled  at  P  (B.  IV., 
P.  Xin.,  S.),  and  consequently,  AP  is  perpendicular  to 
PQ  :  hence,  AP  is  perpendicular  to  every  line  of  the 
plane  MN  passing  through  P,  and  consequently,  to  the 
plane  itself  ;    xoMch  was  to   be  proved. 

11 


162  GEOMETRY. 

Cor.  1.  Only  one  perpendicular  can  be  drawn  to  a  plane 
from  a  point  without  the  plane. 
For,  suppose  two  perpendiculars, 
as  AP  and  AQ^  could  be 
drawn  from  the  point  A  to  the 
plane  MN.  Draw  PQ ;  then 
the  triangle  APQ  would  have 
two  right  angles,  APQ  and 
AQP;    which  is  impossible   (B.  L,  P.  XXV.,  C.  3). 

Cor.  2.  Only  one  perpendicular  can  be  drawn  to  a  plane 
from  a  point  of  that  plane.  For,  suppose  that  two  perpen- 
diculars coixld  be  drawn  to  the  plane  J/iV,  fr'om  the  point 
P.  Pass  a  plane  through  the  perpendiculars,  and  let  PQ 
be  its  intersection  vnih  MN";  then  we  should  have  two  per- 
pendiculars drawn  to  the  same  straight  line  from  a  point  of 
that  line  ;    which  is   impossible    (B.  I.,   P.  XIV.). 


PROPOSITION     V.        THEOREM. 

If  from  a  point  without  a  plane,  a  perpendicular  be  drawn 
to  the  plane,  and  oblique  lines  be  drawn  to  different 
points   of  the  plane  : 

1°.     The  perpendicular  will  be  shorter  than  any  oblique  line  : 

2°.  Oblique  lines  ichich  meet  the  plane  at  equal  distances 
from  the  foot  of  the  perpendicfidar,  will  be  equal : 

8.°  Of  two  oblique  lines  lohich  meet  the  plane  at  unequal 
distances  from  the  foot  of  the  perpendicular,  the  one  which 
meets  it  at  the  greater  distance  will  be  the  longer. 

Let  A  he  a  point  without  the  plane  JfiV ;  let  AP 
be  perpendicular  to  the  plane ;  let  vJ.  (7,  AD,  be  any  two 
oblique  lines  meeting  the  plane  at  equal  distances  from  the 
foot    of  the    perpendicular ;    and  let    AO    and    A£I   be  any 


BOOK     VI. 


163 


two  oblique  lines  meeting  the  plane  at  unequal  distances  from 
the  foot  of  the  jjerpendicular  : 

1°.  AP  will  be  shorter 
than   any   oblique  line    AC. 

For,  draw  PC;  then  wiU 
AP  be  less  than  AC  (B. 
I.,  P.  XV.)  ;  which  was  to 
be  proved.  : 

2®.    AC    and    AD    ynA  be   equal. 

For,  draw  PD  ;  then  the  right-angled  triangles  APC^ 
APD,  will  have  the  side  AP  common,  and  the  sides  PC, 
PP,  equal  :  hence,  the  triangles  are  equal  in  all  their  parts, 
and  consequently,  A  C  and  AP  will  be  equal ;  which  was 
to  be  proved. 

3".    API   will  be  greater  than    AC. 

For,  draw  PP,  and  take  PP  equal  to  PC  ;  draw 
AP  :  then  wiU  AP  be  greater  than  AP  (B.  I.,  P.  XV.)  ; 
but  AP  and  A  C  are  equal :  hence,  AP  is  greater  than 
AC  ;    which  %cas  to  be  proved. 

Cor.  The  equal  oblique  lines  AP,  AC,  AP,  meet  the 
plane  JIfiV  in  the  circumference  of  a  circle,  whose  centre  is 
P,  and  whose  radius  is  PP  :  hence,  to  draw  a  perpendi- 
cular to  a  given  plane  MN,  from  a  point  A,  without  that 
plane,  find  three  points  P,  C,  P,  of  the  plane  equally  dis- 
tant from  A,  and  then  find  the  centre  P,  of  the  circle 
whose  circumference  passes  through  these  points :  then  will 
AP    be   the  perpendicular   required. 

Scholium.  The  angle  APP  is  called  the  inclifiation  of 
the  oblique  line  AP  to  the  plane  MK  The  equal  oblique 
lines  AP,  AC,  AP,  are  all  equally  inclined  to  the  plane 
MN".  The  inclination  of  AP  is  less  than  the  inclination  of 
any  shorter  line    AP. 


164 


G  E  O  jNI  E  T  11  Y. 


PROPOSITION    VI.      THEOREil. 

If  from  the  foot  of  a  perpencUcuIm-  to  a  plane,  a  strai'gJtt  line 
be  drmon  at  right  angles  to  any  straight  line  of  that  pla?ie, 
and  the  point  of  intersection  le  joined  with  any  point  of  the 
^jerpendiculai',  the  last  line  will  be  per2)endicular  to  the  line 
of  the  plane. 

Let  AP  be  perpendicular  to  the  plane  il/iV,  P  its  foot, 
BC  the  given  Ime,  and  A  any  point  of  the  perpendicular; 
draw  PB  at  right  angles  to  BC,  and  join  the  point  B 
with    A  :     then  w411    AB    be   perpendicular   to    B  C. 

For,  lay  off  BB  equal  to 
BC,  and  draw  PB,  PC,  AB, 
and  AC.  Because  PB  is  per- 
pendicular to  BC,  and  BB 
equal  to  BC,  we  have,  PB 
equal  to  PC  (B.  L,  P.  XV.)  ; 
and  because  AP  is  perpendicu- 
lar to  the  plane    MK,     and    PB 

equal  to  PC,  we  have  AB  equal  io  AC  (P.  Y.).  The 
Ime  AB  has,  therefore,  two  of  its  pomts  A  and  B,  each 
equally  distant  from  B  and  C  :  hence,  it  is  perpendicular 
to    BC    (B.  I.,  P.  XVL,  C.)  ;    which  was  to  be  proved. 


Cor.  1.  The  line  BC  is  perpendicular  to  the  plane  of 
the  triangle  APB  ;  because  it  is  perpendicular  to  AB  and 
FB,    at    B    (P.   IV.). 

Cor  2.  The  shortest  distance  between  AP  and  BC  ia 
measured  on  PB,  perpendicular  to  both.  For,  draw  BU 
between  any  other  points  of  the  lines  :  then  will  BJiJ  be 
greater  than  PB,  and  PB  will  be  greater  than  PB  : 
hence,    PB    is  less  than    BK 


BOOK     VI. 


165 


Scholium.  The  lines  AP  and  BCy  though  not  in  the 
same  plane,  are  considered  perpendicular  to  each  other.  In 
general,  any  two  straight  lines  not  in  the  same  plane,  are 
considered  as  making  an  angle  with  each  other,  which  angle 
is  equal  to  that  fonned  by  drawing  through  a  given  point, 
two  lines  respectively  parallel  to  the  given  lines. 


PKOPOsrriox    vn. 


THEOREM. 


If  one  of  Uoo  parallels  is  perpendicular  to  a  plane^  the  othesr 
one  is  also  perpendicular  to   the  same  plar.e. 

Let  AP  and  ED  be  two  parallels,  and  let,  AP  be 
perpendicular  to  the  plane  MN" :  then  will  ED  be  also 
perpendicular   to   the   plane    JLTiV". 

For,  pass  a  plane  through  the 
parallels  ;  its  intersection  with 
Jf/Y  will  be  PD  ;  draw  AD, 
and  in  the  plane  J/iV  draw 
BC  perpendicular  to  PD  at 
D.  Now,  PD  is  perpendicular 
to  the  plane  APDE  (P.  VI.,  0. 1); 

the  angle  BDE  is  consequently  a  right  angle  ;  but  the  an- 
gle EDP  is  a  right  angle,  because  ED  is  parallel  to  AP 
(B.  I.,  P.  XX.,  C.  1)  :  hence,  ED  is  perpendicular  to  BD 
and  PD,  at  their  point  of  intersection,  and  consequently,  to 
their  plane    MN"    (P.  IV.)  ;    which  was   to  he  proved. 

Cor.  1.  If  the  lines  AP  and  ED  are  perpendicular  to 
the  plane  MN",  they  are  parallel  to  each  other.  For,  if 
not,  draw  through  D  a  line  parallel  to  PA  ;  it  will  be 
perpendicular  to  the  plane  J/iV",  from  what  has  just  been 
proved ;  we  shall,  therefore,  have  two  perpendiculars  to  the 
the  plane  M2T,  at  the  same  point ;  which  is  impossible  (P. 
IV.    C.  2). 


166  GEOMETRY. 

Cor.  2.  If  two  straight  lines,  A  and  B,  are  parallel  to  a 
third  line  G,  they  are  parallel  to  each  other.  For,  pass  a 
plane  po^pendicular  to  G\  it  will  be  perpendicular  to  both 
4i   and  B'.    hence,  A  and  B  are  parallel. 


PROPOSITION   VIII.      THEOREM. 

If  a  straight  line  is  parallel  to  a  line  of  a  plane,  it  is  parallel 

to  that  plane. 

Let  the  line    AB     be    parallel  to  the  line    CD     of  the 
plane    MN  \    then  wiU    AB    be    parallel  to  the  plane    MN. 

For,    through     AB     and     CD 
pass  a  plane  (P.  IE.,  C.  4)  ;     CD  ^  g 

will  be  its  intersection  with 
the  plane  MN".  Now,  since  AB 
Ues  in  this  plane,  if  it  can  meet 
the  plane  J/iV,  it  will  be  at 
some  point   of    CD ;    but   this    is 

impossible,  because  AB  and  CD  are  parallel :  hence,  AB 
cannot  meet  the  plane  JlfiV,  and  consequently,  it  is  parallel 
to  it ;    which  was   to  be  proved. 

PROPOSITION     IX.        THEOREBI. 

J[f   two  planes  are  perpendicular    to    the    same    straight  line, 
they  are  parallel  to  each  other. 

Let  the  planes  MN"  and   PQ  ^ ^ 

be  perpendiiular  to  the  line    AB, 

at  the  points    A    and    B  :    then  () 

will    they    be    parallel     to     each 

other. 

For,  if   they  are  not    parallel, 


/  ^ 

r-— /. 

M 

/   ^ 

''7 

^ 

BOOK     VI. 


167 


they  will  meet ;  and  let  0  be  a  point  common  to  both. 
From  0  draw  the  lines  OA  and  OB  :  then,  since  OA 
lies  in  the  plane  MN^  it  will  be  perpendicular  to  BA  at 
A  (D,  1).  For  a  like  reason,  OB  wiU  be  perpendicular 
to  AB  at  B  :  hence,  the  triangle  OAB  will  have  two 
right  angles,  which  is  impossible  ;  consequently,  the  pianos 
cannot  meet,  and  are  therefore  parallel  ;  which  was  to  be 
proved. 


PEOPOSITION      X.        THEOREM. 

If  a  plane  ititersect    two  parallel  plajies^   the    lines  of  inter- 
section will  he  parallel. 

Let  the  plane  JEH  intersect  the  parallel  planes  J/iV  and 
PQ,  in  the  lines  EF  and  GH -.  then  wiU  EF  and  GH 
be  parallel. 

For,  if  they  are  not  parallel, 
they  will  meet  if  sufficiently  pro- 
longed, because  they  lie  in  the 
same  plane ;  but  if  the  lines  meet, 
the  planes  il/iV  and  BQ^  in 
which  they  lie,  will  also  meet  ; 
but  this  is  impossible,  because 
these  planes   are    parallel :    hence, 

the    lines    FF   and    GH    cannot  meet ;    they  are,  therefore, 
parallel  j    which  was  to  he  proved. 


PEOPOSITION     XI.        THEOREM. 

If  a    straight  line    is   perpendicular    to    one   of   two  parallel 
planes^  it  is  also  perpendicular  to  the  other. 

Let  MIT  and  PQ  be  two  parallel  planes,  and  let  the 
lino  AB  be  perpendicular  to  PQ  then  will  it  ako  be 
perpendicular  to    MN". 


168 


GEOMETRY. 


For,   through    AB    pass  any  plane  ;    its  intersections   with 
MN    and    I*Q    will  be  parallel  (P.  X.)  ;    but,  its  intersection 
with    FQ    is    perpendicular    to    AB    at    J?    (D.  1)  ;    hence, 
its    intersection     with      J/iV       is 
also  perpendicular  to    AB    at   A 
(B.    L,    P.   XX.,   C.  1)  :     hence, 
AB     is    perpendicular    to    every 
line   of    the    plane    J/iV    through 
A,     and   is,  therefore,  2)erpendicu- 
lar  to   that  plane  ;    which  was   to 
be  2^oved. 


Q 

/  B 

^/ 

p 

N 

/  ^ 

■^/ 

:m 


PKOPOSITION     Xn.        THEOEEM. 

Parallel  straight  lines  included  ietiueen  parallel  planes,  are  equal 

Let  £JG  and  I^H  be  any  two  parallel  lines  included 
between  the  parallel  planes  J/iV  and  PQ  :  then  wUl  they 
be   equal. 

Through  the  parallels  conceive 
a  plane  to  be  passed  ;  it  will 
intersect  the  plane  Jl/iV  in  the 
line  JEJI]  and  FQ  La  the  line 
G^ ;  and  these  lines  will  be 
parallel  (Prop.  X.).  The  figure 
EFHG  is,  therefore,  a  parallelo- 
gram :  hence,  QE  and  HF 
are  equal   (B.  I.,    P.  XX^nil.)  ;    which  was  to  be  proved. 

Cor.  1.  The  distance  between  two  parallel  planes  is  me!W 
sured  on  a  perpendicular  to  both  ;  but  any  two  perpendiculars 
between  the  planes  are  equal :  hence,  parallel  planes  are  every- 
where  equally  distant. 

Cor.  2.  If  a  straight  line  GH  is  parallel  to  any  plane  MX, 
then  can  a  plane  be  passed  through  GH  parallel  to  MN: 
hence,  if  a  straight  line  is  parallel  to  a  plane,  all  of  its  points 
are  equally  distant  from  that  plane. 


BOOK     VI. 


169 


PROPOSITION     Xni.        THEOKEM 

If  two  angles,  not  situated  in  the  same  plane,  have  their 
sides  parallel  and  lying  in  the  same  direction,  the  angles 
will  be  equal  and  their  planes  parallel. 

Let  CAS  and  DBF  be  two  angles  lying  in  the  planes 
MN  and  PQ,  and  let  the  sides  AG  and  AE  be  re- 
spectively parallel  to  BD  and  BF,  and  lying  in  the  same 
du-ection  :  then  wUl  the  angles  CAF  and  DBF  be  equal, 
and  the  planes    MN    and    FQ    will  be   parallel. 

Take  any  two  points  o^  AG    and    AF,    as    G  and  F,  and 
make  BD     equal  to    A  G,     and 
BF   to    AE',     draw    CE,  BF, 
AB,    CD,    and    EF. 

1°.    The    angles     GAF     and 
DBF   will  be   equal. 

For,  AE  and  BF  bemg 
parallel  and  equal,  the  figure 
ABFF  is  a  paraUelogram  (B. 
I.,  P.  XXX.)  ;  hence,  FF  is 
parallel  and  equal  to  AB.  For 
a  like  reason,  GD  is  parallel  and  equal  to  AB :  hence, 
GD  and  EF  are  parallel  and  equal  to  each  other,  and 
consequently,  GE  and  DF  are  also  parallel  and  equal  to 
each  other.  The  triangles  GAE  and  DBF  have,  therefore, 
their  corresponding  sides  equal,  and  consequently,  the  corres- 
ponding angles  GAF  and  DBF  are  equal ;  which  was  to 
be  proved. 

1".    The  planes  of  the  angles  MN    and  PQ    are  parallel. 

For,  if  not,  pass  a  plane  through  A  parallel  to  PQ, 
and  suppose  it  to  cut  the  lines  GD  and  EF  in  G  and 
H.      Then  will   the    lines     GD    and    HF    be   equal  respect- 


170 


GEOMETRY. 


ively  to  AB  (P.  XII.),  and  consequently,  GD  will  be 
equal  to  CD^  and  JSF  to  -E'jP;  wliich  is  impossible  j  hence, 
the  planes  MIf  and  PQ  must  be  parallel ;  which  was  to 
be  proved. 

Cor.  If  two  parallel  planes  JOT  and  PQj  are  met  by 
two  other  planes  AD  and  AJP]  the  angles  CA£J  and 
DBF^    formed  by  their  intersections,  will  be  equal 


PROPOSITION     XIV.        THEOEEJf. 


If  three  straight  lines^  not  situated   in    the    same  plaiie,  are 

equal  and  parallel,  the    triangles  formed   by  joining    the 

extremities  of  these  lines  will  he    equal,  and  their  planes 
parallel. 

Let  AB,  GD,  and  EF  be  equal  parallel  lines  not  m 
the  same  plane :  then  will  the  triangles  A  CE  and  BDF 
be   equal,   and  their  planes  parallel. 

For,  AB  being  equal  and 
parallel  to  EF,  the  figure  ABFE 
is  a  parallelogram,  and  conse- 
quently, AE  is  equal  and  par- 
allel to  BF.  For  a  like  reason, 
AC  is  equal  and  parallel  to 
BD :  hence,  the  included  angles 
CAE  and  DBF  are  equal  and 
their  planes  parallel  (P.  Xm.). 
Now,  the  triangles  CAE  and 
DBF   have  two  sides  and  their 

included  angles  equal,  each  to  each :  hence,  they  are  equal 
in  all  their  parts.  The  triangles  are,  therefore,  equal  and 
their  planes  parallel ;    which  was  to  be  proved. 


BOOK     VI. 


171 


PROPOSITION     XV.        THEORESI. 

If  two  straight  lines  are  cut    by  three   parallel   planes^   they 
will  be  divided  proportionally. 

Let  tho  lines  AB  and  CD  be  cut  by  the  parallel 
Planes  MN,  PQ,  and  RS^  in  the  points  A^  JEl,  2?,  and 
C,    F,    D\    then 

AE    '.    EB    w    CF  ',    FD. 

For,  draw  the  line  AD^  and 
suppose  it  to  pierce  the  plane 
PQ  in  G\  draw  AC,  BB, 
EG,    and    GF. 

The  plane  ABB  intersects 
the  parallel  planes  PS  and  PQ 
in  the  lines  BB  and  EG  ; 
consequently,  these  lines  are  par- 
aUel  (P.  X.)  :  hence  (B.  IV., 
P.  XV.), 

AE   :    EB    '.'.AG    :     GB. 

The  plane  AGB  intersects  the  parallel  planes  MN  and 
PQ,     in  the  parallel  lines    AC    and    GF :     hence, 

AG    '.     GB  :  :     CF   :    FB. 

Combining  these  proportions  (B.  IT.,  P.  IV.),  we  have, 

AE   '.    EB  '.'.     CF   '.    FB  \ 

which  was  to  be  proved. 

Cor.  1.  If  two  straight  lines  are  cut  by  any  number  of 
parallel  planes,   they  will  be  divided  proportionally. 

Cor.  2.  If  any  number  of  straight  lines  are  cut  by  three 
parallel  planes,  they  will  be  divided  proportionally. 


172  GEOMETRY. 

PKOPOSITION     XVI.        THEOREM. 

Tf  a  straiglit  line  is  perpe?idicular  to  a  pla7ie,  every  plane  passed 
throngli  the  line  will  also  be  2}er2)endiciclar  to  that  j;?awe. 

Let  AF  be  perpendicular  to  the  plane  MN',  and  let 
JiF  be  a  plane  passed  through  AP  :  then  will  BF  bo 
perpendicular  to    J/iV. 

In    the    plane     J/iV,      draw     FD 
perpendicular    to     FC,     the    intersec- 
tion of    FF   and    J/iV.      Since    AF  r 
is  perpendicular  to     il/iVj      it    is  per-            / 

pendicular   to   i^C   and    DF    (D.  1);  /     ^'    ^ 22/ 

and    since     AF     and      Z>P,     in    the 

planes  BF  and  MN^  are  perpendicular  to  the  intersection 
of  these  planes  at  the  same  point,  the  angle  which  they 
form  is  equal  to  the  angle  formed  by  the  planes  (D.  4)  ; 
but  this  angle  is  a  right  angle  :  hence,  BF  is  perpendicu- 
lar to    MN ;    which  was  to  be  proved. 

Cor.  If  three  lines .  AF,  BF,  and  DF,  are  perpen- 
dicular to  each  other  at  a  common  point  F,  each  line  will 
be  pei'pendicular  to  the  plane  of  the  other  two,  and  the 
three   planes  will   be   perpendicular   to   each  other. 

PROPOSITION     XVn.        THEOREM. 

If  two  planes  are  perpendicular  to  each  other,  a  straight  line 
draiun  in  one  of  them,  p)crpendicular  to  their  intersection, 
will  he  perpendicular   to  the  other. 

Let  the  planes  BF  and  J/iV  be  perpendicular  to  each 
other,  and  let  the  line  AF,  drawn  in  the  plane  BF,  be 
perpendicular  to  the  intersection  BC  ;  then  will  AF  be 
perpendicular   to   the   plane    MN. 


BOOK     VI 


173 


For,  in  the  plane  il/iVj  draw  PB  perpendicular  to  BQ 
at  P.  Then  because  the  planes  BF  and  MN  are  perpen- 
dicular to  each  other,  the  angle  APB 
"will  be  a  right  angle  :  hence,  AP  is 
perpendicular  to  the  two  lines  PB 
and  i>C,  at  their  intersection,  and 
consequently,  is  perpendicular  to  their 
plane    MN  \    which  was  to   he  proved. 

Cor.  If  the  plane  BF  is  perpendicular  to  the  plane 
JIfiV,  and  if  at  a  point  P  of  their  intersection,  we  erect 
a  perpendicular  to  the  }^\^ne  JIJSTy  that  perpendicular  will 
be  in  the  plane  BF.  For,  if  not,  draw  in  the  plane  BF, 
PA  perpendicular  to  PC,  the  common  intersection  ;  AP 
will  be  perpendicular  to  the  plane  J/iV,  by  the  theorem ; 
therefore,  at  the  same  point  P,  there  are  two  perpendiculars 
to    the    plane     il/iV ;     which   is  impossible   (P.  IV.,   C.  2). 


PROPOSITION      XVin.        THEOREM. 

If  two  planes  cut  each  other,  and  are  perpendicular  to  a 
third  pkme,  their  i?itersection  is  also  perpendicular  to 
that  plane. 

Let    the    planes     BF,     BJT,    be    perpendicular  to     ifOT : 
tten   will   their   intersection    AP    be   perpendicular  to    iJOT. 

For,  at  the  point  P,  erect  a  per- 
pendicular to  the  plane  JT/iV ;  that 
perpendicular  must  be  in  the  plane 
BF,  and  also  in  the  plane  BS" 
(P.  XVn.,  C.)  ;  therefore,  it  is  their 
common  intersection  AP;  which  was 
to  be  proved. 


174  GEOMETRY. 

PROPOSITION     XIX.        THEOEEM. 

The    sum    of  any  two    of   the   plane  angles  formed  by  tM 
edges  of  a  triedral  angle,  is  greater  than   the  third. 

Let  SA-f  SBy  and  SO,  be  the  edges  of  a  triedral 
angle  :  then  will  the  sum  of  any  two  of  the  plane  angles 
formed  by  them,  as  ASG  and  CSB,  be  greater  than  the 
third    A  SB. 

If  the  plane  angle  A  SB  is  equal  to,  or  less  than,  either 
of  the  other  two,  the  truth  of  the  proposition  is  evident.  Let 
us  suppose,  then,  that    ASB    is  greater  than   either. 

Li  the    plane     ASB,     construct 
the  angle    BSD    equal    to    BSC  ; 
draw    AB    in  that  plane,    at    plea- 
sure ;     lay  off    SG     equal  to    SB, 
and    draw     AG     and     GB.      The 
triangles     BSD     and     BSG    have 
the    side     SG     equal  to     SD,    by 
construction,     the     side     SB      com- 
mon,  and    the  included    angles    BSD    and    BSG     equal,  by 
construction  ;    the    triangles    are  therefore    equal    in  all  theii 
parts  :    hence,    BD    is  equal  to  BG.      But,  from  Proposition 
Vn.,   Book  L,  we  have, 

BG  +  GAy  BD  +  DA. 

Taking  away  the  equal  parts    BG    and    BD,    we  have, 

GA  >  DA; 

hence   (B.  L,  P.  IX.),  we  have, 

angle    ASG  >  angle    ASD  ; 

and,   adding  the   equal  angles    BSG    and    BSD, 


BOOK     VI.  175 

angle  ASC  +  angle  CSB  >  angle  ASD  +  angle  DS.B  ; 
or,  angle  ASG  +  angle  CSJB  >  angle  ASH  ; 

which  was  to  be  proved. 


PEOPOSITION     XX.        THEOEEM. 

The  sum  of  the  plane  angles  formed  by  the  edges  of  any 
polyedral  angle^   is  less  than  four  right  angles. 

Let  S  be  the  vertex  of  any  polyedral  angle  whose  edges 
are  SA^  SB,  SC,  SB,  and  S-E ;  then  will  the  sum  of 
the   angles   about    S    be  less  than   four  right   angles. 

For,  pass  a  plane  cutting  the  edges 
La  the  points  A,  B,  C,  B,  and  E, 
and  the  faces  in  the  lines  AB,  BC, 
CB,  BE,  and  EA.  From  any  point 
within  the  polygon  thus  formed,  as  0, 
draw  the  straight  lines  OA,  OB,  OC, 
OB,     and    OE. 

"We  then  have  two  sets  of  triangles, 
one  set  having  a  common  vertex  S,  the 
other  havuig  a  common  vertex  0,  and  both  having  com- 
mon bases  AB,  BC,  CB,  BE,  EA.  Now,  in  the  set 
which  has  the  common  vertex  S,  the  sum  of  all  the  angles 
is  equal  to  the  sum  of  all  the  plane  angles  formed  by  the 
edges  of  the  polyedral  angle  whose  vertex  is  S,  together 
with  the  sura  of  all  the  angles  at  the  bases  ;  viz.,  SAB, 
SB  A,  SBC,  &c.  ;  and  the  entire  sum  is  equal  to  twice 
as  many  right  angles  as  there  are  triangles.  In  the  set 
whose  common  vertex  is  0,  the  sum  of  all  the  angles  is 
equal  to  the  four  riglit  angles  about  0,  together  with  the 
interior  angles  of  the  polygon,  and  this  sum  is  equal  to 
twice    as  many   right    angles    as    there    are    triangles.      Since 


176 


GEOMETRY. 


the  nnmber  of  triangles,  in  eacli  set,  is  the  same,  it  follows 
that  these  sums  are  equal.  But  in  the  triedral  angle  whose 
vertex   is    J5,     we  have    (P.  XLX.), 

S 
ABS  +  SBC  >  ABC  ; 

And    the    like    may  be    shown    at    each 

of   the    otlier  vertices,    C,    Z>,    ^,     A  : 

hence,    the    sum    of    the    angles   at-  the 

bases,   in    the    triangles   whose    common 

vertex   is    S,     is    greater  than    the   sum 

of   the   angles    at    the  bases,  in   the   set 

whose  common  vertex  is    0 :     therefore, 

the   sum   of  the   vertical    angles    about     S^      is    less    than    the 

sum    of   the    angles    about     0  :     that   is,   less  than   four  right 

angles  ;    which  was   to  he  proved. 

Scholium.  The  above  demonstration  is  made  on  the  sup- 
position that  the  polyedral  angle  is  convex,  that  is,  that  the 
diedral  angles  of  the  consecutive  faces  are  each  less  than  two 
right  angles. 


PROPOSITION     XXT.        THEOEEM. 

If  the  plane  angles  formed  by  the  edges  of  two  triedral 
angles  are  eqnal,  each  to  eac\  the  planes  of  the  equal 
angles  are  equally  inclined  to  each  other. 

Let  S  and  T  be  the  vertices  of  two  triedral  angles, 
and  let  the  angle  ASC  \>^  equal  to  DTF,  A  SB  to  DTE, 
and  BSC  to  ETF :  then  will  the  planes  of  the  equal 
angles  be   equally  inclined  to   each   other. 

For,  take  any  point  of  SB,  as  B,  and  from  it  draw 
in  the  two  faces  ASB  and  CSB,  the  lines  BA  and  BC, 
respectively  perpendicular  to  SB  :  then  will  the  angle  ABC 
measure    the  inclination  of  these  faces.      Lay  off    TE    equal 


BOOK     VI, 


177 


to  SB,  and  from  B  draw  in  the  faces  DTH  and  FTBy 
the  lines  ED  and  BF,  respectively  perpendicular  to  TB - 
then  will  the  angle  DBF 
measure  the  inclination  of  these 
faces.  Draw  AC  and  BF. 
The  right-angled  triangles 
S:BA  and  TBD,  have  the 
side  SB  equal  to  TB,  and 
the     angle     ASB      equal     to 

BTB ',  hence,  AB  is  equal  to  BB,  and  AS  to  Ti?. 
In  hke  manner,  it  may  be  shown  that  BC  is  equal  to  BF, 
and  CS  to  FT.  The  triangles  ^/5C  and  BTF,  have 
the  angle  ^aS'C  equal  to  BTF,  by  hypothesis,  the  side  AS 
equal  to  BT,  and  the  side  CaS^  to  FT,  from  what  has 
just  been  shown ;  hence,  the  triangles  are  equal  in  all  their 
parts,  and  consequently,  AO  is  equal  to  BF.  Now,  the 
triangles  ABC  and  BBF  have  their  sides  equal,  each  to 
each,  and  consequently,  the  corresponding  angles  are  also 
equal;  that  is,  the  angle  ABC  is  equal  to  BBF:  hence, 
the  inclination  of  the  planes  ASB  and  CSB,  is  equal  to 
the  inclination  of  the  planes  BTB  and  FTB.  In  like 
manner,  it  may  be  shown  that  the  planes  of  the  other  equal 
angles   are    equally  inclined  ;    which  was   to   be  proved. 

Scholium.  If  the  planes  of  the  equal  plane  angles  are 
like  placed,  the  triedral  angles  are  equal  in  all  respects,  for 
they  may  be  placed  so  as  to  coincide.  If  the  planes  of  the 
equal  angles  are  not  similarly  placed,  the  triedral  angJes  are 
equal  by  symmetry.  In  this  case,  they  may  be  placed  so 
that  two  of  the  homologous  facos  shall  coincide,  the  triedral 
angles  lying  on  opposite  sides  of  the  plane,  which  is  then 
called  a  plane  of  symmetry.  In  this  position,  for  every  point 
on  one  side  of  the  plane  of  symmetry,  there  is  a  correspond- 
ing point   on  the    other   side. 


BOOK  VII. 


POLTBDBONS. 


DEFINITIONS. 


1.  A  PoLTEDROX  is  a  volume  bounded  by  polygons. 

The  bounding  pol^'gons  are  called  faces  of  the  polycdron ; 
the  lines  in  wliicli  the  faces  meet,  are  called  edges  of  the 
polyedron  ;  the  points  in  which  the  edges  meet,  are  called 
vertices   of  the  polyedron. 

2.  A  Prism  is  a  polyedron  in  which  two  of 
the  faces  are  polygons  equal  in  all  their  parts, 

and  having  their  homologous  sides  parallel.    The 

other  faces  are  parallelograms  (B.  I.,  P.  XXX.). 

The  equal  polygons  are  called  bases  of  the 

prism ;     one    the     upper^    and     the    other    the 

lower  base ;    the   parallelograms  taken   together 

make    up    the    lateral    or  convex    surface  of    the   prisrn 

lines  in  which  the  lateral  faces  meet,  are   called  lateral 

of  the   prism. 


;    the 

edges 


3.    The    Altitudh    of    a    prism    is    the    perpendicular    dis- 


tance between   the  planes   of  its   bases. 

4.  A  IliGUT  Prism  is  one  whose  lateral 
edges  are  perpendicular  to  the  planes  of  the 
bases. 

In  this  case,  any  lateral  edge  ia  equal  to 
the   altitude. 


B>i 


'  T' 


BOOK     VII. 


179 


5.     An    Obliqub    Prisir    is    one   whose    Litornl    el;^r'3    are 
oblique  to   the  planes   of  the   bases. 

In   this  case,  any  lateral   edge  is  greater  tlian  the  altitude. 


6.  Prisma  are  named  from  the  number  of  sides  of  their 
bases  ;  a  triangular  prism  is  one  whose  bases  are  triangles ; 
u  pentangular  prism  is  one  whose  bases  are  pentagons,   &c. 

7.    A    PARALLELOPiPEDO]!ir    13    a    prism    whose    bases    are 
parallelograms. 

A  Right  rarallelopipedon  is  one  whose  lat- 
eral edges  are  perpendicular  to  the  planes 
of  the  bases. 

A  Rectangular  ParaUelojjipedon  is  one 
whose  faces  are  all  rectans-les. 


\ 


K 


A  Cube  is  a  rectangular  j)arallelopipedon 
whose  faces   are  squares. 

8.  A  PrRAinD  is  a  polyedron  bounded 
by  a  polygon  called  the  base,  and  by  tri- 
angles meeting  at  a  common  point,  called  the 
vertex    of  the   pyramid. 

The  triangles  taken  together  make  up  the 
lateral  or  convex  surface  of  the  pyramid  ; 
the  lines  in  which  the  lateral  faces  meet,  are 
called  the   lateral  edges    of  the  pyramid. 


9.  Pyramids  are  named  from  the  number  of  sides  of 
their  bases ;  a  triangidar  pyramid  is  one  whose  base  is  a 
tnangle ;  a  quadrangular  pyramid  is  one  whose  base  is  a 
quadiilatcral,   and   so   on. 

10.  The    Altitude    of    a    pyramid    is    the    perpendicular 
distance  from  the  vertex   to  the  plane  of  its  base. 


180  GEOMETRY. 

11.  A  Right  Pyramid  ia  one  whose  base  is  a  regular 
polygon,  and  in  which  the  perpendicular  drawn  from  the 
vertex  to  the  plane  of  the  base,  passes  through  the  centre 
of  the   base. 

This  perpendicular  is   called  the   axis   of  the   pyramid. 

12  The  Slajstt  Height  of  a  right  pjTamid,  is  the  per- 
pendicular distance   from   the  vertex   to   any   side   of  the   base. 

13.  A  TKirN"CATED  Pyramid  is  that 
portion  of  a  pjTamid  included  between 
the  base  and  any  plane  which  cuts  the 
{)yTaraid. 

When   the   cutting   plane    is  parallel   to 
the   base,   the   truncated  pyramid  is   called 
a  FRUSTUM   OF   A  PY'RAMiD,    and   the   inter- 
section  of  the   cutting   plane   with  the    pyramid,   is   called    tlie 
ujyper  base  of  the  frustum  ;    the   base   of   the   pyi-amid   is  call- 
ed   the  lower  base   of  the   frustum. 

14.  The  Altitude  of  a  frustum  of  a  pyi-amid,  is  the  per- 
pendicular  distance   between   the   planes  of  its  bases. 

15.  The  Slant  Height  of  a  frustum  of  a  right  pyramid, 
ia  that  portion  of  the  slant  height  of  the  pyramid  which  lies 
between   the   planes   of  its   upper   and   lower  bases. 

16.  Similar  Poltedkoxs  are  those  which  are  bouiided  by 
the   same  number   of  similar  polygons,   similarly  placed. 

Parts  which  are  similarly  placed,  whether  faces,  edges,  or 
angles,  are   called  homologous. 

17.  A  Diagonal  of  a  polyedron,  is  a  straight  line  join- 
ing the  vertices  of  two  polyedral  angles  not  in  the  same 
face. 


BOOK     VII. 


181 


18.  The  Volume  of  a  Polyedeon  is  its  numerical  value 
expressed   in   terms   of  some   other  polyedron   as   a   unit. 

The  unit  generally  employed  is  a  cube  constructed  on  the 
linear  unit  as  an  edge. 


PKOPOSITION     I.        THEOREM. 


The  convex  surface  of  a  right  prism  is  equal  to  the  perim- 
eter of  either  base  multiiylied  by  the  altitude. 

Let    AB  CDE-K    be  a  right  prism  :    then  is  its  convex 
surface   equal  to, 

i^AB  ^  BC  +  CD  +  BE  +  EA)   x  AF. 

For,  the  convex  surface  is  equal  to 
the  sum  of  all  the  rectangles  A  (r,  BH^ 
CI^  BK,  EF^  which  compose  it.  Now, 
the  altitude  of  each  of  the  rectangles 
AF,  BG,  CH,  &c.,  is  equal  to  the 
altitude  of  the  prLsm,  and  the  area  of 
each  rectangle  is  equal  to  its  base  mul- 
tiphed  by  its  altitude  (B.  IV.,  P.  V.)  : 
hence,  the  sum  of  these  rectangles,  or 
the   convex   surface   of  the  prism,   is    equal   to, 

{AB  +  BC  +  CD  +  BE+  EA)   x  AF ; 

that  is,   to   the  perimeter   of   the  base   multiplied  by  the  alti- 
tude ;  which  was  to  be  2^oved. 

Cor.  If  two  right  prisms  have  the  same  altitude,  theii" 
convex  surfaces  are  to  each  other  as  the  peruneters  of  their 
bases. 


182 


GEOMETRY. 


PROPOSITION   II.      THEOREM. 

In  any  prism,  the  sections  made  hj  parallel  planes  are  polygons 

equal  in   all  their  j^ar/s. 

Let  the  prism  AH  be  intersected  by  the  parallel  planes 
NT,  SV :  then  are  the  sections  NOPQR,  STVXY, 
equal   polygons. 

For,  the  sides  NO,  ST,  are  parallel, 
being  the  intersections  of  parallel  planes 
with  a  third  plane  ABGF;  these  sides, 
NO,  ST,  are  included  between  the  par- 
allels NS,  OT:  hence,  NO  is  equal  to 
ST  (B.  L,  P.  XX\TIL,  C.  2).  For  Uke 
reasons,  the  sides  OP,  PQ,  QP,  &c., 
of  NOPQP,  are  equal  to  the  sides 
TV,  VX,  &c.,  of  STVXY,  each  to 
each  ;  and  since  the  equal  sides  are  par- 
allel,  each    to    each,    it   follows    that    the 

angles  NOP,  OPQ,  &c.,  of  the  first  section,  are  equal  to 
the  angles  STV,  TVX,  &c.,  of  the  second  section,  each  to 
each  (B.  VI.,  P.  XIII.)  :  hence,  the  two  sections  NOPQP, 
STVXY,  are  equal  in  all  their  parts;  ^uhich  loas  to  he  proved. 

Cor.  The  bases  of  a  prism,  and  every  section  of  a  prism, 
parallel  to  the  bases,   are  equal  in  all  their  parts. 


PROPOSITIOISr     III.        THEOREM. 

If  a  pyramid  he  cut  hy  a  plane  parallel  to   the  hase  • 

1°.    The  edges  and  the  altitude  will  he  divided  p>roportionally : 
2".    The  section  will  he  a  polygon  similar  to   the  hase. 

Let    the    pyramid     S-ABCDE,     whose    altitude    is      SO, 
be    cut   by  the  plane    ahcde,     parallel  to  the  base    ABODE. 


BOOK     VII. 


183 


l".    The   edges   and  altitude  will  be  divided  proiioilionally. 

Foi,  conceive  a  plane  to  be   passed  through  the  vei'tex  /S, 
parallel    to    the    jDlane    of    the   base  ;     thou 
will    the    edges    and     the    altitude     be   cut  S 

by  three  parallel  planes,  and  consequently 
they  will  be  divided  proportionally  (B.  VI., 
P.  XV.,   C.  2)  ;    which  was   to   he  lyroved. 


2°.  The  section  ahcde^  will  be  similar 
to  the  base  ABODE.  For,  ah  is  par- 
allel to  AB,  and  he  to  BC  (B.  VI., 
P.  X.)  :  hence,  the  angle  ahc  is  equal  to 
the  angle  ABC.  In  like  manner,  it  may 
be  shown  that  each  angle  of  the  polygon  ahcde  is 
to  the  corresponding  angle  of  the  base  :  hence,  the 
polygons   are   mutually   equiangular. 

Agam,   because    ah    is   parallel   to    AB,     we   have, 


equal 
two 


ah 


AB 


sh 


SB  ; 


and,   because     he    is   parallel   to    BC,     we  have, 

ho    :     BC     :  :     sh    :     SB  ; 
hence    (B.  II.,    P.  IV.),   we   have, 

ah    :     AB    :  .     he    :     BC. 

In   like   manner,    it    may   be    shown   that    all  the    sides   of 

abede     are    proportional    to    the    corresponding  sides     of   the 

polygon    ABCJDJS  :     hence,    the   section     ahede  is  similar  to 

the  base    ABCBE  (B.  IV.,  D.  1)  ;    xohich  teas  to  oe  lyroved. 

Cor.  1.  If  two  pyramids  S- ABC  BE,  and  S-XYZ, 
ha\'ing  a  common  vertex  S,  and  their  bases  in  the  same 
plane,  be  cut  by  a  f)lane  ahc,  parallel  to  the  plane  of 
their   bases,   the   sections   will   be   to    each   other   as   the   bases. 


184 


GEOMETRY. 


For,  tlie  polygons  ahcd  and  ABCD,  being  similar,  arc 
to  each  other  as  the  squares  of  their  homologous  sides  ah 
•dad    J  J]    (B.  IV.,   P.  XXVn)  ;    but, 


aP    :     AB""    :  :    Set      :     SA 
Lence   (B,  11.,  P.  IV.),   we  have, 


abcde   \  ABCDE 


ISo^   :  80^. 


In   hke   manner,    we   have, 

xyz    :    XYZ   :  ;     So"    :     JO^  ;     ^ 

hence,  '  3 

abcde    :     ABODE    :  :    xyz    :    XYZ. 


Cor.   2.     If  the   bases   are   equal,  any  sections   at   equal  dis- 
tances from  the   bases  will   be   equal. 

Cor.  3.     The   area   of   any   section    parallel   to   the  base,    is 
proportional   to   the   square   of  its  distance   from  the  vertex. 


PROPOSITION     IV. 


THEOREM. 


The    convex    surface    of   a    right    pyramid    is    equal    to    tJie 
perimeter  of  its   base  multiplied  by  half  the  slant  height. 

Let     S     be    the    vertex,     ABCDE  the 

base,   and     SE,     perpendicular   to     EA,  the 

slant   height   of    a    right    pyramid  :    then  will 
the   convex   surface   be   equal  to, 

{AB  +  BC+  CD  +  DE+  EA)    x  \SF. 

Draw    SO    perpendicular  to   the   plane   of  the 
base. 


BOOK     VII.  185 

From  the  definition  of  a  right  pyramid,  the  point  0  is 
the  centre  of  the  base  (D.  11)  :  hence,  the  lateral  edges, 
SA,  S.B,  etc.,  are  all  equal  (B.  VI.,  P.  V.)  ;  hut  the  sides 
of  the  base  are  all  equal,  being  sides  of  a  regular  polygon  : 
hence,  the  lateral  faces  are  all  equal,  aud  consequently  their 
altitudes  are  all  equal,  each  being  equal  to  the  slant  heiglit 
of  the   pyramid. 

Now,  the  area  of  any  lateral  face,  as  SEA,  is  equal  to 
its  base  EA,  multij^lied  by  half  its  altitude  SF :  hence, 
the  sura  of  the  areas  of  the  lateral  faces,  or  the  convex  sui-- 
face   of  the  pyramid,   is   equal  to, 

{AB  +  J5C  -h  CD  +  BE  +  EA)  x  \SF ; 

which  was  to  he  proved. 

Scholium.  The  convex  surface  of  a  fi-ustum  of  a  right 
pyramid  is  equal  to  half  the  sum  of  the  perimeters  of  its 
upper  and  lower  bases,  muUijjlied  hy  the  slant  hei(jht. 


Let  ABCDE-e  be  a  frustum  of  a  right 
pjTamid,  whose  vertex  is  S  :  then  will  the 
section  ahcde  be  similar  to  the  h^^o,  ABODE, 
and  their  homologous  sides  will  be  parallel, 
(P.  III.).  Any  lateral  face  of  the  frustum, 
as  AEea,  is  a  trapezoid,  whose  altitude  is 
equal  to  Ff  the  slant  height  of  the  frustum  ; 
hence,  its  area  is  equal  to  \{EA  +  e«)  X  Ff 
(B.  IV.,  P.  VII.).  But  the  area  of  the  con- 
vex surface  of  the  frustum  is  equal  to  the  sum  of  the  areas 
of  its  lateral  faces ;  it  is,  therefore,  equal  to  the  half  sura 
•  )f  the  perimeters  of  its  upper  and  lower  bases,  multiplied 
by  the  slant  heiglit. 


I8G 


GEOMETRY. 


PKOPOSITIOX    V.      THEOREM. 


If  ilic  three  faces  wMcli  include  a  iriedral  angle  of  a  prism 
are  eqiial  in  all  their  2}a'>'ts  to  the  thj'ee  faces  which  include 
a  iriedral  angle  of  a  second  prisni,  each  to  each,  and  are 
like  placed,   the  tico  2^^'isms  are  equal  in  all  their  i)arts. 

i.et  Z>  aud  h  be  the  vertices  of  two  triedral  angles, 
mcluded  by  faces  respectively  equal  to  each  other,  aud  simi- 
larly placed  :  then  will  the  prism  ABQDE-K  be  equal  to 
the  i^rism    abcde-h^    in  all   of  its  parts. 

For,  place  the  base 
abode  uj^on  the  equal 
base  ABC  BE,  so  that 
they  shall  coincide  ;  then 
because  the  triedral  an- 
gles whose  vertices  are 
b  and  jB,  are  equal, 
the  parallelogram  hh  will 
coincide  with  DII,  and 
the  parallelogram  bf  "with 
SF :     hence,    the     two 

sides  fg  and  gh,  of  one  upper  base,  will  comcide  with  the 
homologous  sides  of  the  other  upper  base  ;  and  because  the 
upper  bases  are  equal  iu  all  their  parts,  they  must  coincide 
throughout;  consequently,  each  of  the  lateral  faces  of  one 
prism  will  coincide  with  the  corresponding  lateral  face  of  tlie 
other  prism  :  the  prisms,  therefore,  coincide  throughout,  and 
ere  therefore  equal  in  all  their  parts ;   ^vh^ch  was  to  he  proved. 

Cor.  If  two  right  prisms  have  their  bases  equal  in  all  their 
parts,  aud  have  also  equal  altitudes,  the  prisms  themselves  will 
be  equal  in  all  their  parts.  For,  the  faces  which  include  any 
triedral  angle  of  the  one,  will  be  equal  in  all  their  parts  to 
the  faces  which  include  the  corresponding  triedral  angle  of 
the  other,  each  to  each,  and  they  will  be  similarly  placed. 


BOOK     VII.  187 


PEOPOSITION     YI.        THEOREM. 

In  any  pai'aUelojjijJcclon,  the  opposile  faces  are  equal  in  all  their 
parts,  each  to  each,  and  their  pla7ies  are  parallel. 

Let     ABCD-H    be    a    parallolopii^edon  :     then    will    its 
opposite   tiices   be   equal   and  tlieir  planes   will  be   parallel. 

For,  the  bases,  ABCB  and  EFGR 
are  equal,  and  tbelr  planes  parallel  by 
definition  (D.  '7).  The  opposite  faces 
AEHD  and  BFGC,  have  the  sides  AE 
and  BF  parallel,  because  they  are  oppo- 
site sides  of  the  parallelogram  BE ; 
and    the    sides     EH    and     EG    parallel, 

because  they  are  opposite  sides  of  the  parallelogram  EG ; 
and  consequently,  the  angles  AEII  and  BEG  are  equal 
(B.  M:.,  p.  XIII.).  But  the  side  AE  is  equal  to  BF,  and 
the  side  EII  to  FG  ;  hence,  the  faces  AEHD  and 
BFGC  are  equal  ;  and  because  AE  is  parallel  to  BF, 
and  EII  to  FG,  the  planes  of  the  faces  are  parallel 
(B.  V^I.,  P.  Xni.).  In  like  manner,  it  may  be  shown  that 
the  parallelograms  ABEE  and  BCGH,  are  equal  and  their 
planes  parallel  :  hence,  the  opposite  faces  are  equal,  each  to 
each,  and  their  planes   are  parallel  ;    which  was  to   he  proved. 

Cor.   1.     Any    two    opposite    faces    of    a    parallelopipedon 
may  be   taken   as   bases. 

Cor.  2.  In  a  rectangular  parallelo- 
pipedon, the  square  of  either  of  the 
diagonals  is  equal  to  the  sum  of  the 
squares  of  the  three  edges  which  meet 
at  the   same   vertex. 

For,  let    FD    be   either  of  the   diagonals,    and  draw    FS. 


B 


188 


GEOMETRY. 


Then,    in    the    right-angled    triangle    FIID^    we  have, 


But    DH   is    equal   to    FB,      and    FH^ 
is   equal  to    FA^    plus    AU^    or     FU^  : 


hence, 


FJf  =  FB^  +  FA^  +  FC\ 


Cor.  3.  A  iiarallelopipedon  may  be  constructed  on  three 
straight  lines  AB,  AD,  and  AE,  intersecting  in  a  common 
point  A,  and  not  lying  in  the  same  plane.  For,  pass  through 
the  extremity  of  each  line,  a  plane  parallel  to  the  plane  of 
the  other  two;  then  will  these  planes,  together  with  the 
planes  of  the  given  lines,  be  the  faces  of  a  parallelopipedon. 


PROPOSITION     VII. 


TUEOREM. 


If  a  plane  he  passed  through  the  diagonally  opposite  edges 
of  a  parallelopipedon^  it  loill  divide  the  parallelopipedon 
into   two   equal  triatigidar  pyrisms. 

Let    ABCB-H     be    a     parallelopipedon,   and   let   a   plane 
be   passed  through  the  edges    BF    and    DH  •     then  will  the 
prisms    ABB-H  and    BCB-H  be   equal 
in  volume. 

For,  through  the  vertices  F  and  B 
iet  planes  be  passed  perpendicular  to 
FB.,  the  former  cutting  the  other  lateral 
edges  in  the  points  6,  A,  g,  and  the 
latter  cutting  those  edges  produced,  in 
the  points  a,  <?,  and  c.  The  sections 
Fehg    and    Bade    will  be  parallelograms, 


BOOK     YII.  189 

because  tlieir  opposite  sides  are  parallel,  each  to  each  (B.  VI., 
P.  X.)  ;  they  vdW  also  be  equal  (P,  II.)  :  hence,  the  poly- 
edron  Badc-g  is  a  right  prism  {D.  2,  4),  as  are  also  the 
polyedrons    Bad-h    and    Bcd-h. 

Place  the  triangle  Feh  upon  Bad,  so  that  F  shall 
coincide  with  B,  e  with  «,  and  h  with  d  ;  then, 
because  eEy  hBT,  are  perpendicular  to  the  plane  Fc/i,  and 
aA,  dB,  to  the  plane  Bad,  the  line  eF  Avill  take  the 
direction  a  A,  and  the  line  hU  the  direction  dJ).  The 
lines  AF  and  ae  are  equal,  because  each  is  equal  to  BF 
(B.  I.,  P.  XXVni.).  If  we  take  away  from  the  line  aE 
the  part  ae,  there  will  remain  the  part  eE  ;  and  if  fi-om 
the  same  line,  we  take  away  the  part  AE,  there  will  re- 
main the  part  Aa  :  hence,  eE  and  aA  are  equal  (A.  3)  ; 
for  a  like  reason  hll  is  equal  to  dB :  hence,  the  point 
E  will  coincide  with  A,  ami  the  point  H  with  B,  and 
consequently,  the  polyedrons  Feh-H  and  Bad-B  will 
coincide   throughout,   and   are   therefore   equal. 

If  from  the  polyedron  Bad -II,  we  take  away  the 
part  Bad-B,  there  will  remain  the  prism  BAB-II ; 
and  if  from  the  same  polyedron  we  take  away  the  part 
Feh-H,  there  will  remain  the  prism  Bad-h  :  hence, 
these  prisms  are  equal  in  volume.  In  like  manner,  it  may 
be  shown  that  the  prisms  BCB-H  and  Bcd-h  are  equal 
in   volume. 

The  prisms  Bad-h,  and  Bcd-h,  have  equal  hases,  be- 
cause these  bases  are  halves  of  equal  parallelograms  (B.  I., 
P.  XXVIIL,  C.  1)  ;  they  have  also  equal  altitudes  ;  they  are 
therefore  equal  (P.  V.,  C.)  :  hence,  the  prisms  BAB-II  and 
BCB-H    are   equal   (A.  1)  ;    which   was   to   be  proved. 

Cor.  Any  triangular  prism  ABB-II,  is  equal  to  half  of 
the  parallelopipedon  AG,  which  has  the  same  triedral  angle 
4,     and   the  same    edges    AB,    AB,     and    AE. 


190 


GEOMETRY. 


PROPOSITION     Vni.        THEOREM. 

If  two  parallelopipedons  have  a  common  loioer  hase^  and 
their  upper  bases  beticeen  the  same  parallels^  they  are 
equal  in  volume. 

Let  the  parallelopipedons  AG  and  AL  have  the  com- 
mon lower  base  ABCD,  and  their  upper  bases  EFGII 
and  IKLM^  between  the  same  parallels  EK  and  IIL  : 
then  will   they  be   equal  in   volume. 

For,  the  lines  EF  and 
IK  are  equal,  because  each 
is  equal  to  AB  ;  hence, 
the  sum  of  EF  and  FI^ 
or  EI^  is  equal  to  the 
Bum  of  FI  and  IK^  or 
FK.  In  the  triangular 
prisma  AEI-M         and 

BFK-L,  we  have  the  line  AE  equal  and  parallel  to 
BE,  and  ET  equal  to  FK ;  hence,  the  face  AEI  is 
equal  to  BFK.  In  the  faces  EIMII  and  FKLG,  we  liavo, 
HE=.GF,  Er=FK  and  HEI—GFK:  hence,  the  two  faces 
arc  e(i;ial  (Bk.  I.  T.  xxviii.  G.  3)  :  the  faces  AEHD  and  BFGC 
are    also    equal     (P.    VI.)  :     hence,    the    prisms    arc    equal     (P. 

V.) 

If    from    the    polyedron     ABKE-H,    we    take    away   the 

prism    BFK-L^    there  will   remain  the  parallelopipedon    A  G  j 

and   if   from    the    same    polyedron   we    take    away   the    prism 

AEI-M,     there  will  remain  the  parallelopipedon   AL :    hence, 

these    parallelopipedons    are    equal    in   volume    (A.  3)  ;    which 

was  to  be  proved. 


BOOK     VII. 


191 


PKOPOSITION     IX.        THEOREM. 

If  two  paralleloplpedons  have  a  common  loioer  base  and  tJie 
same  altitude^   they  will  be    equal  in  volume. 

T^et  the  parallelopipedons  AG  and  AL  have  the  com- 
mou  lower  base  ABCD  and  the  same  altitude:  then  will 
they   be   equal  in   volume. 

Because  they  have  the  same  altitude,  their  upper  bases 
will  lie  in  the  same  plane. 
Let  the  sides  IJSI  and  KL 
be  prolonged,  and  also  the 
sides  FE  and  GH ;  these 
prolongations  will  form  a 
parallelogram  0  Q,  which 
will  be  equal  to  the  com- 
mon base  of  the  given  par- 
allelopipedons, because  its 
sides  are  respectively  parallel 
and  equal  to  the  correspond- 
ing  sides   of  that   base. 

Now,  if  a  third  parallelopipedon  be  constructed,  having 
for  its  lower  base  the  parallelogram  AJ3CD,  and  for  its 
upper  base  NOPQ^  this  third  parallelopipedon  will  be  equal 
in  volume  to  the  parallelopipedon  AG^  since  they  have  the 
same  lower  base,  and  their  upper  bases  between  the  same 
parallels,  QG^  KF  (P.  Vm.).  For  a  like  reason,  this 
third  parallelopipedon  will  also  be  equal  in  volume  to  the 
parallelopipedon  AL  :  hence,  the  two  parallelopipedons  AG 
AL^     are   equal   in  volume  ;    which  was   to  be  proved. 

Cor.  Any  oblique  parallelopipedon  may  be  changed  into  a 
right  parallelopipedon  having  tlie  same  base  and  tlie  same 
altitude ;    and  they  will  be  equal  in  volume. 


192 


GEOMETRY. 


PKOPOSITION      X.        PROBLEM. 


To  construct  a  rectangular  parallelojnpedon  lohich  shall  be 
equal  in  volume  to  a  right  'paralleloinpedon  whose  base 
is   any  parallelogram. 

Let    ABGD-M    be    a    right    parallelopipedon,   having    for 
its  base   the   parallelogram    ABCD. 

Through   the   edges  AI    and    JBIT  pass 
the    planes     A  Q      and     BP^      respectively 
perpendicular  to    the    plane    AIT^     the   for- 
mer meeting  the    face     BL    in     OQ^     and 
the   latter   meeting   that    face     produced  in 
NP :    then   will    the   polyedron    AP    be   a 
rectangular    parallelopipedon     equal    to    the 
given   parallelopipedon.       It  will  be   a  rect- 
angular  parallelopipedon,   because   all    of  its 
faces    are    rectangles,    and    it    will    be    equal     to    the    given 
parallelopipedon,  because   the   two   may  be  regarded  as  having 
the  common   base    AK  (P.  VI.,  C.  1),  and   an   equal   altitude 
AO  (P.  IX.). 


Cor.  1.  Since  any  oblique  parallelopipedon  may  be  changed 
into  a  right  parallelopipedon,  having  the  same  base  and  alti- 
tude, (P.  IX.,  Cor.) ;  it  follows,  that  any  oblique  parallelopipedon 
may  be  changed  into  a  rectangular  parallelopipedon,  having 
Au   equal  base,  an   equal  altitude,  and  an  equal  volume. 

Cor.  2.  An  oblique  parallelopipedon  is  equal  in  volume  to 
a  rectangular  parallelopipedon,  having  an  equal  base  and  an 
equal   altitude. 

Cor.  3.  Any  two  parallelopipedons  are  equal  in  volume 
when  they  have  equal  bases  and  equal  altitudes. 


BOOK      VII. 


193 


PROPOSITION     XI.        THEOREM. 


Thco    rectangular  parallelopipedons    having  a    common    lower 
basCf  are  to  each  other  as  their  altitudes. 

Let  the  parallelopipedons  AG  and  Al>  have  the  .com 
mon  lower  base  ABC  J):  then  will  they  be  to  each  other 
as  their   altitudes    AE    and    AI. 

1°.  Let  the  altitudes  be  commensurable,  and  suppose,  for 
example,   that    AE    is  to    AI,     as    15    is  to    8. 

Conceive  AE  to  be  divided  into  15  equal  parts,  of 
which  AI  will  contain  8  ;  through  the  points  of  division 
let  planes  be  passed  parallel  to  ABCD.  These  planes  will 
divide  the  parallelopipedon  AG  into  15  parallelopipedons, 
which  have  equal  bases  (P.  11.  C.)  and  equal  altitudes ; 
hence,   they   are   equal   (P.  X.,  Cor.  3). 

Now,  AG  contains  15,  and  AL  8 
of  these  equal  parallelopipedons  ;  hence, 
AG  is  to  AI,  as  15  is  to  8,  or  as 
AE  is  to  AI.  In  like  manner,  it  may 
be  shown  that  AG  is  to  AI,  as  AE 
is  to  AI,  when  the  altitudes  are  to  eaoh 
other   as  any   other   whole   numbers. 

2°.    Let  the   altitudes  be    incommensur- 
able. 

Now,  '■£  AG  \a  not  to  AI,  as  AE  is  to  -47,  let  U6 
suppose   that, 

AG    '.    AI    :  '.    AE   :    AO, 


in   which    AO    is  greater  than    AI. 

Divide    AE    into    equal    parts,    such    that    each    shall    be 
less  than     01 ;    there   will   be   at   least   one   pomt   of   division 

13 


194 


GEOMETRY. 


wi,  between  0  and  I.  Let  P  denote  the  parallelopipe- 
don,  whose  base  is  ABCD^  and  altitude  Am  ;  since  the 
altitudes  AE^  Am^  are  to  each  other  as  two  whole  num- 
bers,  \\'e  have, 


AG    '.    P    '.  :    AE   :    Am. 

But,   by  hypothesis,   we    have, 

AG    :    AL    :  :    AE    :    AO; 
therefore   (B.  H.,  P.  IV.,  C), 

AZ    :    P    :  :    AO    :    Am. 


But  AO  is  greater  than  Am  ;  hence,  if 
the  proportion  is  true,  APi  must  be  greater  than  P.  On 
the  contrary,  it  is  less  ;  consequently,  the  fourth  term  of 
the  proportion  cannot  be  greater  than  AT.  In  like  manner, 
it  may  be  shown  that  the  fourth  term  cannot  be  less  than 
AI ;  it  is,  therefore,  equal  to  AT.  In  this  case,  therefore, 
AG    is  to    -4Z,    as    AE   is  to    AI. 

Hence,  in    all    cases,    the    given    parallelopipedons    are    to 
each   other  as  their  altitudes  ;    which  was   to  be  proved. 

Sch.    Any  two  rectangular  parallelopipedons  whose  bases  are 
equal  in  all   their  parts,  are  to  each  other  as  their  altitudes. 


PKOPOSITION     Xn.        THEOEEM. 

7^0  rectangular  parallelopipedons  having  equal  altitudes^  are 
to  each  other  as  their  bases. 

Let  the  rectangular  parallelopipedons  AG  and  AE^  have 
the  same  altitude  AE :  then  will  they  be  to  each  other  as 
their  bases. 


BOOK     VII 


195 


K 


M 


N 


^ 


vol.  A  G    :    vol.  A  Q 


AB 


B 
AO. 


a 


V3 


For,  place  them  as  shown  in  the  figure,  and  produce  the 
plane  of  the  face  iVX,  until 
it  intersects  the  plane  of  the 
face  ITC,  in  PQ ;  we  shall 
thus  form  a  third  rectangular 
parallelopipedon    A  Q, 

The  parallelopipedons  AG 
and  A  Q  have  a  common 
base  A£r ;  they  are  there- 
fore to  each  other  as  their 
altitudes  AB  and  AO 
(P.  XI.)  :  hence,  we  have 
the  proportion, 


The  parallelopipedons  AQ  and  A^  have  the  common  base 
AL ;  they  are  therefore  to  each  other  as  their  altitudes 
AD    and    AM :    hence, 


vol.  AQ    :    vol.  AK 


AD 


AM. 


Multiplying  these  proportions,  term  by  term   (B.  11.,  P.  XII.), 
and  omitting  the  common  factor,    vol.  A  Qy    we  have, 


vol.  AG    :    vol.  AK 


AB  X  AD    :    AO  X  AM. 


But    AB  X  AD    is  equal  to  the  area  of  the  base    ABCD' 
and    AO  X  A3f  is   equal   to  the  area  of  the  base    AMNO 
hence,    two    rectangular    parallelopipedons    having    equal    alti 
tudes,   are    to    each    other  as  their  bases ;    which   was   to   he 
proved. 


196 


GEOMETRY. 


PROPOSITION     Xni.        THEOEEM. 

Any  two  rectangular  parallelopi2yedons  are  to  each  other  as 
the  products  of  their  bases  and  altitudes  ;  that  is,  as  the 
products  of  their  three  dimensions. 

Let    AZ    and    AG    be 

any  two  rectangular  paral- 
lelopipedons :  then  will  they 
be  to  each  other  as  the 
products  of  their  three  di- 
mensions. 

For,  place  them  as  in  the 
6gure,  and  produce  the  faces 
necessary  to  complete  the 
rectangular  parallelopipedon 
AJy.  The  parallelopipedons 
AZ  and  AK  have  a  com- 
mon   base     AJS^ ;    hence    (P.  XI.), 

vol  AZ    :    vol  AK   :  :    AX    :    AM 

The    parallelopipedons     AK   and     AG     have    a    common 
altitude    AB ;    hence   (P.  XII.), 

volAK   :    vol  AG    ::    A3IN0    :    ABCD. 

Multiplying    these    proportions,   term    by    term,   and    omitting 
the  common  factor,    vol  AK^    we  have, 

^lAZ    :    vol  AG    :  :    AMJ^O  x  AX    :    ABCD  x  AB; 

or,  since    A3IN0   is  equal  to    A3I  x  AO,    and  AB  CD    to 
AB  X  AD, 

volAZ  :  vol  AG   :  :  A3fxA0xAX  :  AB  x  AB  kAB-, 

which  was  to  he  proved. 


BOOK     VII. 


197 


Cor.  1.    If  we  make    the    tliree    edges     A31^     AO,  and 

A^,    eacli   equal  to   tlie  linear  unit,  the   parallelopipedon  AZ 

will  be  a   cube   constructed   on   that   unit,   as    an    edge  ;  and 

consequently,    it    will    be    the    unit    of    volume.      Under  this 
supposition,   the  last  proportion    becomes. 


whence, 


vol.  AG     :  :     1     :     AB  x  AD  x  AE  \ 
vol  AG   =   AB  X    AB  x  AK 


Hence,  the  volume  of  any  rectangular  parallelopipedon  m 
equal  to  the  product  of  its  three  dimensions  ;  that  is,  tho 
number  of  times  which  it  contains  the  unit  of  volume,  is 
equal  to  the  number  of  linear  units  in  its  length,  by  the 
number  of  linear  units  in  its  breadth,  by  the  number  of 
linear   units  in  its  height. 

Cor.  2.  The  volume  of  a  rectangular  parallelopipedon  is 
equal  to  the  product  of  its  base  and  altitude  /  that  is,  the 
number  of  times  which  it  contains  the  unit  of  volume,  is 
equal  to  the  number  of  superficial  units  in  its  base,  multi- 
plied  by  the   number   of  linear  units   in   its   altitude. 

Cor.  3.  The  volume  of  any  parallelopipedon  is  equal  to 
the   product   of  its  base   and   altitude    (P.  X.,  C.  2). 


i 


PROPOSITION      XIV.        THEOREM. 

The  volume  of  any  prism  is    equal    to    the   product    of  its 

base  and  altitude. 

Let  ABCDE-K  be  any  prism  :  then  is  its  volume 
equal  to   the  product   of  its  base   and   altitude. 

For,  through  any  lateral  edge,  as  AF,  and  the  other  lateral 
edges  not  in  the  same  faces,  pass  the  planes  AH,  AI,  dividing 
the  prism  into  triangular  prisms.  These  prisms  will  all  have 
a   common  altitude   equal   to  that  of  the  given  prism. 


198 


GEOMETRY. 


Now,  the  volume  of  any  one  of  the  triangular  prisms,  as 
ABG-R,  is  equal  to  half  that  of  a  parallelopipedon  con- 
structed on  the  edges  BA,  BC^  BG 
(P.  VII.,  C.)  ;  but  the  volume  of  this  par- 
allelopipedon is  equal  to  the  product  of  its 
l)ase  and  altitude  (P.  XIII.,  C.  3)  ;  and 
because  the  base  of  the  prism  is  half 
that  of  the  parallelopipedon,  the  volume 
of  the  prism  is  also  equal  to  the  pro- 
duct of  its  base  and  altitude :  hence, 
the  sum   of  the  triangular   prisms,   which 

make  up  the  given  prism,  is  equal  to  the  sum  of  their 
bases,  which  make  up  the  base  of  the  given  prism,  mto 
their  common  altitude  ;    which  was  to  he  proved. 

Cor.  Any  two  prisms  are  to  each  other  as  the  products 
of  their  bases  and  altitudes.  Prisms  having  equal  bases  are 
to  each  other  as  their  altitudes.  Prisms  having  equal  alti- 
tudes are  to   each  other  as  their  bases. 


PEOPOSITION     XV.        THEOKEM. 


Two  triangular  x>yramids  having  equal  bases  and  equal  alti- 
tudes^  are  equal  in  volume. 

Let  S-ABG,  and  S-ahc^  be  two  pyramids  having  their 
equal  bases  ABG  and  aha  in  the  same  plane,  and  let  A2' 
be  their  common  altitude  :  then  will  they  be  equal  in  vol- 
ume. 

For,  if  they  are  not  equal  in  volume,  suppose  one  of 
them,  as  S-ABG,  to  be  the  greater,  and  let  their  differ- 
ence be  equal  to  a  prism  whose  base  is  ABG^  and  whose 
altitude  is    Aa. 


BOOK     VII 


199 


Divide  the  altitude  AT  into  equal  parts  Ax^  jcy,  &c., 
each  of  which  is  less  than  Aa^  and  let  k  denote  one  of 
these  parts  ;  through  the  points  of  division  pass  planes  par- 
allel to  the  plane  of  the  bases  ;  the  sections  of  the  two 
pyramids,  by  each  of  these  planes,  will  be  equal,  namely, 
DEF   to    clef,     GRI    to    ghi,    &c.    (P.  HI.,  C.  2). 


u- 


On  the  triangles  ABC,  DEF,  &c.,  as  lower  bases,  con- 
struct exterior  prisms  Avhose  lateral  edges  shall  be  parallel 
to  AS,  and  whose  altitudes  shall  be  equal  to  h\  and  on  the 
triangles  def,  glii,  &c.,  taken  as  upper  bases,  construct  inte- 
rior prisms,  whose  lateral  edges  shall  be  parallel  to  Sa,  and 
whose  altitudes  shall  be  equal  to  h.  It  is  evident  that  the 
sura  of  the  exterior  prisms  is  greater  than  the  pyi-amid 
8- ABC,  and  also  that  the  sum  of  the  interior  prisms  is  less 
than  the  pyramid  S-ahc  :  hence,  the  difference  between  the 
sum  of  the  exterior  and  the  sura  of  the  interior  prisms,  is 
greater  than  the   difference   between   the  two   pyramids. 

Now,  beginning  at  the  bases,  the  second  exterior 
prism    EFD-G,     is   equal   to   the    first   interior  prism     efd-cu 


200 


GEOMETRY. 


because  they  have  the  same  altitude  ^,  and  their  bases 
EFD,  efd^  are  equal :  for  a  like  reason,  tJtie  third  exterior 
prism  HIG-K^  and  the  second  interior  prism  hig-d^  are 
equal,  and  so  on  to  the  last  in  each  set :  hence,  each  of  the 
exterior  prisms,  excepting  the  first  BCA-D^  has  an  equal 
corresponding  interior  prism  ;  the  prism  BGA-D^  is,  there- 
fore, the  diiFerence  between  the  sum  of  all  the  exterior 
prisms,  and  the  sum  of  all  the  interior  prisms.  But  tlie 
difierence  between  these  two  sets  of  prisms  is  greater  than 
that  between  the  two  pyramids,  which  latter  difierence  was 
supposed  to  be  equal  to  a  prism  Avhose  base  is  BOA,  and 
whose  altitude  is  equal  to  Aa^  greater  than  k  ;  conse- 
quently, the  prism  BCA-J)  is  greater  than  a  prism  having 
the  same  base  and  a  greater  altitude,  which  is  im23ossible  . 
hence,  the  supposed  inequality  between  the  two  pyramids 
cannot  exist  ;  they  are,  therefore,  equal  in  volume ;  which 
was  to  be  proved. 


PROPOSITION      XYI. 


TnEORE^I. 


Any  triangidar  prism  inay  he   divided  into   three   triangular 
pyramids,   equal  to  each  other  in  volum,e. 


Let  ABC-B  be  a  triangular 
prism  :  then  can  it  be  divided  into 
three   equal   triangular   pyramids. 

For,  through  the  edge  ACy 
pass  the  plane  ACF,  and  through 
the  edge  EF  pass  the  plane 
EFC.  The  pyramids  AGF-F  and 
ECD-F,  have  their  bases  ACE 
and  ECD  equal,  because  they  are 
halves  of  the  same  parallelogram 
ACDE\    and  they  have  a  common 


BOOK     VII. 


201 


altitude,  because  tlieir  bases  are  in  the  same  plane  AD,  and 
their  vertices  at  the  same  point  F ;  hence,  they  are  equal 
in  volume  (P.  XV.)-  The  pyramids  ABC-F  and  DEF-C, 
have  their  bases  ABC  and  DEF^  equal  because  they  are 
the  bases  of  the  given  prism,  and  their  altitudes  are  equal 
because  each  is  equal  to  the  altitude  of  the  prism  ;  they 
ire,  therefore,  equal  in  volume :  hence,  the  three  pyramids 
into  which  the  prism  is  divided,  are  all  equal  in  volume  ; 
jihich  was   to   be  2>^'ovecl. 

Cor.   1.     A    triangular    pyramid    is    one-third    of    a    prism, 
having  an   equal   base   and   an   equal  altitude. 

Cor.   2.     The   volume   of   a   triangular   pyramid   is   equal   to 
one-third   of  the  product   of  its   base   and   altitude. 


PROPOSITION     XVn.        THEOREM. 

The  volume    of   any   pyramid   is    equal  to   one-third  of  the 
product   of  its   base   and  altitude. 

Let    S-ABCDE,     be    any  pyramid:    tlien    is    its   volume 
equal   to   one-third   of  the   product   of  its  base   and   altitude. 

For,  through  any  lateral  edge,  as  SE, 
pass  the  planes  SEB.,  SEC,  dividing  the 
pyramid  into  triangular  pyramids.  The  alti- 
tudes of  these  pyramids  Avill  be  equal  to 
each  other,  because  each  is  equal  to  that 
f-f  the  given  pyi-amid.  Now,  the  volume 
;i?  each  triangular  pyramid  is  equal  to  one- 
third  of  the  product  of  its  base  and  alti- 
tude (P.  XVI.,  C.  2)  ;  hence,  the  sum  of 
the  volumes  of  the  triangular  pyramids,  is 
equal   to   one-third   of  the   product   of   the   sum  of  their   ba^c^; 


202  GEOMETRY. 

by  their  common  altitude.  But  the  sum  of  the  triangular 
pyramids  is  equal  to  the  given  pyramid,  and  the  sum  of 
their  hases  is  equal  to  the  base  of  the  given  pyramid : 
hence,  the  volume  of  the  given  pyramid  is  equal  to  one- 
thiid  of  the  product  of  its  base  and  altitude  j  which  was  to 
he  proved. 

Cor.  1.  The  volume  of  a  pyramid  is  equal  to  one-third 
of  the  volume  of  a  prism  having  an  equal  base  and  an  equal 
altitude. 

Cor.  2.  Any  two  pyramids  are  to  each  other  as  the 
products  of  their  bases  and  altitudes.  Pyramids  having  equal 
bases  are  to  each  other  as  their  altitudes.  Pyramids  having 
equal  altitudes  are  to   each  other  as  their  bases. 

Scholium,  The  volume  of  a  polyedron  may  be  found  by 
dividing  it  into  triangular  pyramids,  and  computing  their 
volumes  separately.  The  sum  of  these  volumes  will  be  equal 
to  the  volume   of  the  jDolyedron. 


PROPOSITION     XVin.        THEOREM. 

TJie  volume  of  a  frustum  of  any  triangular  pyramid  is 
equal  to  the  sum,  of  the  volumes  of  three  jyyramids 
whose  common  altitude  is  that  of  the  frustum^  and  whose 
bases  are  the  lower  base  of  the  frustum^  the  upper  base 
of  the  frustum^  and  a  mean  proportional  between  the  two 
bases. 

Let  FGII-h  be  a  fi  ustum  of  any  triangular  pyramid  : 
then  will  its  volume  be  equal  to  that  of  three  pyramids 
whose  common  altitude  is  that  of  the  frustum,  and  whose 
bases  ave  the  lower  base  FGH^  the  upper  base  fgh^  and 
a   mean   proportional  between   their   bases. 


BOOK     VII. 


203 


For,   tlirougL   the   edge    FH^    pass  the    plane    I^Hg^     and 
through    the    edge    fg^     pass    tlie    plane    fyll.,     dividing  the 
frustum  into  thi*ea  pyi-ainids.      The  pyra- 
mid   g-FGII^    has  for  its  base  the  lower 
base    FGH    of  the  frustum,    and   its   al- 
Jtude   is   equal   to   that    of   the    frustum, 
because  its  vertex    </,    is  in  the  plane  of 
he    upper  base.      The    pyramid    H-fgh, 
Las  for  its  base  the  upper  base  fgh    of 
the   frustum,  and  its  altitude  is  equal   to 
that   of    the    frustum,  because  its   vertex 
lies  in   the  plane   of  the   lower   base. 

The  remaining  pyramid  may  be  regarded  as  having  the 
triangle  FfH  for  its  base,  and  the  point  g  for  its  vertex. 
From  <7,  draw  gK  parallel  to  fF^  and  draw  also  KII  and 
Kf.  Then  will  the  jDyramids  K-FfH  and  g-FfH,  be  equal; 
for  they  have  a  common  base,  and  their  altitudes  are  equal, 
because  their  vertices  JT  and  g  are  in  a  line  parallel  to 
the   base    (B.  VI.,  P.  Xn.,  C.  2). 

Now,  the  pyramid  K-FfS  may  be  regarded  as  having 
FKH  for  its  base  and  /  for  its  vertex.  From  K^  draw 
KL  parallel  to  GH ;  it  will  be  parallel  to  gh  :  then  will 
the  triangle  FF^Z  be  equal  to  fgh,  for  the  side  FF  is 
equal  to  fg,  the  angle  F  to  the  angle  /,  and  the  angle  F 
to  the  angle  g.  But,  FKH  is  a  mean  ])roportional  between 
FKL  and  FGH  (B.  IV.,  P.  XXIV.,  C),  or  between  fgh 
and  FGH.  The  pyi-amid  f-FKH,  has,  therefore,  for  its 
base  a  mean  proportional  between  the  upper  and  lower  bases 
of  the  frustum,  and  its  altitude  is  equal  to  that  of  the  frus- 
tum ;  but  the  pyramid  f-FKH  is  equal  in  volume  to  the 
pyramid  g-FfH-.  hence,  the  volume  of  the  given  frustum  is 
equal  to  that  of  three  pyi'amids  whose  common  altitude  is 
equal  to  that  of  the  frustum,  and  whose  bases  are  the  upper 
base,  the  lower  base,  and  a  mean  proportional  between 
them  ;    which  was   to   be  proved. 


204: 


GEOMETRY. 


Cor.  The  volume  of  the  frustum  of  any  pyramid  is 
equal  to  the  sum  of  the  volumes  of  three  pyramids  whose 
common  altitude  is  that  of  the  frustum,  and  xohose  bases 
are  t/ie  lower  base  of  the  frustum,  the  ripper  base  of  the 
frustum,   and  a  mean  j^roj^ortioyial  between   them. 

For,  let  ABCBE-e  be  a  frustum  of 
any  pyramid.  Through  any  lateral  edge,  as 
eE,  pass  the  planes  eEBb,  eECc,  divid- 
ing it  into  triangular  frustums.  Now,  the 
sum  of  the  volumes  of  the  triangular  frus- 
tums is  equal  to  the  sum  of  three  sets  of 
pyramids,  whose  common  altitude  is  that  of 
the  given  frustum.  The  bases  of  the  first 
set  make  up  the  lower  base  of  the  given 
frustum,  the  bases  of  the  second  set  make  up  the  upper  base 
of  the  given  frustum,  and  the  bases  of  the  third  set  make 
up  a  mean  proportional  between  the  upper  and  lower  base 
of  the  given  frustum :  hence,  the  sum  of  the  volumes  of 
the  first  set  is  equal  to  that  of  a  pyramid  whose  altitude  is 
that  of  the  frustum,  and  whose  base  is  the  lower  base  of 
of  the  frustum  ;  the  sum  of  tlie  volumes  of  the  second  set 
is  equal  to  that  of  a  pyramid  whose  altitude  is  that  of  the 
frustum,  and  whose  base  is  the  upper  base  of  the  frustum  ; 
and,  the  sum  of  the  third  set  is  equal  to  that  of  a  pyra- 
mid whose  altitude  is  that  of  the  frustum,  and  whose  base 
is  a  mean  proportional   between   the   two   bases. 


PROPOSITION      XIX. 


THEOREM. 


Similar  triangular  prisms  are   to   each   other  as   the  cubes  af 

their  homologous  edges. 

Let  CBD-P,  obd-p,  be  two  similar  triangular  prisms, 
and  let  B  C,  be,  be  any  two  homologous  edges :  then  will 
the  prism    CBD-P   be  to  the  prism   cbd~p,    as   BO     to    be 


BOOK     VII, 


205 


For,  the  homologous  angles  B  and  b  are  equal,  and 
the  faces  which  bound  them  are  similar  (D.  IG)  :  hence, 
these  tricdral  angles  may  bo 
applied,  one  to  the  other,  so 
that  the  an 2:1  e  chd  will  coin- 
cide  with  CBD^  the  edge  ha 
with  BA.  In  this  case,  the 
prism  cbd-p  will  take  the 
position  Bcd-p.  From  A 
draw     AH     perpendicular     to 

the  common  base  of  the  prisms  :  then  will  the  plane  BAH 
be  perpendicular  to  the  plane  of  the  common  base  (B.  VI., 
P.    XVL).  From       a^       in     the    plane     BAJJ^      draw     ah 

perpendicular  to  BU :  then  will  ah  also  be  perpendicular 
to  the  base  BBC  (B.  VI.,  P.  XVII.)  ;  and  AH,  ah,  will 
be  the   altitudes   of  the   two   prisms. 

Since  the  bases    GBB,    chd,    are  similar,  we  have   (B.  IV., 
P.  XXV.), 


base  CBB 


base  chd 


CB' 


cb  . 


Now,  because  of  the  similar  triangles  ABU,  aBh^  and  of 
the   similar   parallelograms    A  C,    ac,    we   have, 

AH   '.ah     :  :     CB    :     cb  \ 
hence,   multiplying  these   proportions  term  by  term,  we  have, 

base  CBB  x  AH   :    base  chd  x  ah    :  :     GB     :     cb  . 

But,  base  CBB  y^  AH  is  equal  to  the  volume  of  the  prism 
CDB-A,  and  base  chd  X  ah  is  equal  to  the  volume  of 
the    prism     cbd-p ;     hence, 

Xrrism  CBB-P    :     prism  chd-p     :  :      CB^     :     cb  ; 
which  was  to  be  proved. 


206 


GEOMETRY. 


Cor.  1.  ^Iwy  tioo  similar  prisms  are  to  each  other  as 
the  cubes   of  their  homologous   edges. 

For,  since  the  prisms  are  similar,  their  bases  are  similar 
polygons  (D.  16)  ;  and  these  similar  polygons  may  each  be 
divided  into  the  same  number  of  similar  triangles,  similarly 
placed  (B.  IV.,  P.  XXVI.)  ;  therefore,  each  prism  may  be 
divided  into  the  same  number  of  triangular  prisms,  having 
their  faces  similar  and  like  placed  ;  consequently,  the  tri- 
angular prisms  are  similar  (D.  16).  But  these  triangular 
prisms  are  to  each  other  as  the  cubes  of  their  homologous 
edges,  and  being  like  parts  of  the  polygonal  prisms,  the 
polygonal  prisms  themselves  are  to  each  other  as  the  cubes 
of  their   homologous   edges. 

Cor.  2.  Similar  prisms  are  to  each  other  as  the  cubes 
of  their  altitudes,  or  as  the  cubes  of  any  other  homologous 
lines. 

PEOPOSITIOTT      XX.        THEOREM. 


Sim,ilar   pyramids    are    to    each    other  as  the  cubes  of  their 

homologous  edges. 

Let     S-ABCDE,     and     S-abcde^     be    two    simDar  pyra> 
mids,   so    placed    that  their  homologous   angles   at  the   vertex 
shall    coincide,   and   let    AH     and    ab     be 
any  two  homologous  edges  :    then   will  the 
pyramids   be   to    each   other    as    the    cubes 
of   AB    and    ab. 

For,  the  face  SAB,  being  similar  to 
Sab,  the  edge  AB  is  parallel  to  the 
edge  ab,  and  the  face  SBC  being  simi- 
lar to  Sbc,  the  edge  BC  is  parallel  to 
be  ;  hence,  the  planes  of  the  bases  are 
parallel    (B.  VI.,  P.  Xm.). 


BOOK     VII. 


< 

V 


207 


Draw    SO    perpendicular  to   the    base    ABODE  \    it   will 
also   be  perpendicular  to   the   base    ahcde.      Let  it  pierce  that 
plane    at    the    point     o  :      then    will     SO 
be  to    So^    as     SA     is  to    Sa     (P.   III.),  S 

or  as    AB    is   to     ab ;    hence, 

\S0    :    iSo    :  :    AB    :    ab. 

But   the   bases    being   similar   polygons,   we 

have   (B.  IV.,  P.  XXVH.),  A< 

base  AB  CBE    :    base  abode    :  :    Alf    :     ab^. 


Multiplying  these   proportions,   term  by  term,   we   have, 

base  ABODE  X  \S0    :    base  abode  X  \So    ::    AB^    :    ab\ 

But,  base  ABODE  x  ^SO  is  equal  to  the  volume  of  the 
pyramid  S- ABODE,  and  base  abode  X  \So  is  equal  to 
the  volume  of  the  pyramid    S-abcde ;    hence. 


pyramid  S-AB  ODE    :    pyramid  S-abcde    :  :     AB      •    ab  ; 


which  was  to  be  proved. 


Cor.     Similar  pyramids  are  to  each  other  as  the  cubes   of 
their   altitudes,    or    as    the    cubes    of   any    other    homologous 


lines. 


208  GEOMETRY. 


GENERAL       FORMTJLAS. 

If  we  denote  the  volume  of  any  prism  by  "F,  its  base 
by     B^    and  its  altitude  by    IT^    we   shall   have    (P.  XIV.), 

V  =  B  X  II (1.) 

If  we  denote  the  volume  of  any  pyramid  by  F",  its 
base  by    B^    and   its  altitude   by    H^     we  have    (P.  XVII.), 

V  =  \B   X   H (2.) 

If  we  denote  the  volume  of  the  frustum  of  any  pyramid 
by  V^  its  lower  base  by  -Z?,  its  uj^per  base  by  5,  and 
its  altitude  by    H,    we  shall  have    (P.   XVHI.,  C), 


V  =  {{B  +  b  +  y/B   X   b)    X   H    '     •     (3.) 

REGULAR   POLTEDRONS. 

A  Eegular  Polyedron"  is  one  whose  faces  are  all  equal 
regular  polygous,  and  whose  polyedral  angles  are  equal. 
each  to  each. 

There   are   five   regular   polyedrons,  namely  : 

1.  The  Teteaedron,  or  regular  pyramid — a  polyedron 
bounded   by  four   equal   equilateral   triangles. 

2.  The  Hexaedron,  or  cube — a  polyedron  bounded  by 
six    equal   squares. 

3.  The  OcTAEDRON — a  polyedron  bounded  by  eight  equal 
equilateral   triangles. 

4.    Tlie    DoDECAEDRON — a    polyedron    bounded    by    twelve 
equal   and   regular  pentagons. 


BOOK     VII.  209 

5.  The  IcosAEDnoN — a  polyedron  bounded  by  twenty 
equal   equilateral  triangles. 

In  the  Tetraedrou,  the  triangles  are  grouped  about  the 
polyedral  angles  in  sets,  of  three,  in  the  Octaedron  tliey  are 
grouped  in  sets  of  four,  and  in  the  Icosaedron  they  are 
grouped  in  sets  of  five.  Now,  a  greater  number  of  equi- 
lateral triangles  cannot  be  grouped  so  as  to  form  a  salient 
polyedral  angle ;  for,  if  they  could,  the  sum  of  the  plane 
angles  formed  by  the  edges  would  be  equal  to,  or  greater 
than,   four   right  angles,   which   is  impossible    (B.  VI.,  P.  XX.). 

In  the  Hexaedron,  the  squares  are  grouped  about  the 
polyedral  angles  in  sets  of  three.  Now,  a  greater  number 
of  squares  cannot  be  grouped  so  as  to  forai  a  salient  polye- 
dral  angle  ;    for  the   same   reason   as  before. 

In  the  Dodecaedron,  the  regular  pentagons  are  grouped 
about  the  polyedi*al  angles  in  sets  of  three,  and  for  the  same 
reason  as  before,  they  cannot  be  grouped  in  any  greater 
number,   so   as  to   form   a  saUent  polyedral   angle. 

Furthermore,  no  other  regular  polygons  can  be  grouped 
eo   as  to   form   a   saHent   polyedral   angle  ;    therefore, 

Oiily  Jive  regular   polyedrons   can   be   formed. 

U 


BOOK  yiii. 


THB   CTLINDEE,   THE   CONK,   AND   THE   SPHEEB. 


DEFI]SnTIO:N^S. 


1.  A  Cylinder  is  a  volume  which  may  be  generated  by 
a  rectanofle   revolvins:   about   one   of  its   sides  as   an   axis. 

Tlius,  if  the  rectangle  ABCD  be  turned  about  the  side 
AB^    as   an   axis,   it   will   generate  the  cylinder    FGCQ-P. 

The  fixed  line  AB  is  called  the  axis 
of  the  cylinder  /  the  curved  surface  generated 
by  the  side  C-Z),  opposite  the  axis,  is  called 
the  co7iv€x  surface  of  the  cylinder ;  the  equal 
circles  FGCQ^  and  EHDP^  generated  by 
the  remaining  sides  BG  and  AD.,  are  called 
bases  of  the  cylinder  /  and  the  perpendicular 
distance  between  the  planes  of  the  bases,  is 
called   the   altitude   of  the  cylinder. 

The  line  Z>(7,  which  generates  the  convex  surface,  is,  in 
any  position,  called  an  element  of  the  surface  /  the  elements 
are  all  perpendicular  to  the  planes  of  the  bases,  and  any 
one   of  them  is   equal  to   the   altitude   of  the   cylinder. 

Any  line   of   the    generating    rectangle    ABCD.,      as     IK., 
which    is    perpendicular   to    the    axis,   will    generate    a    circle 
whose   plane   is   perpendicular  to   the   axis,  and  which  is  equa 
to   either  base  :    hence,   any  section   of  a   cylinder  by   a   plan 
perpendicular  to    the    axis,   is   a  circle   equal    to    either   base 
Any   sectiort,    FCDE.,     made    by   a    plane    through    the   axis 
is  a  rectangle   double   the   generating  rectangle. 


BOOK     VIII. 


211 


2.  Similar  Cylinders  are  those  wliich  ni.ay  be  generated 
by   similar   rectangles   revolving   about   homologous   sides. 

The  axes  of  similar  cylinders  are  projDortional  to  the  radii 
of  their  bases  (B.  IV.,  D.  1)  ;  they  are  also  proportional  to 
any   other  homologous   lines   of  the   cylinders. 


3.  A  prism  is  said  to  be  inscribed 
in  a  cylinder,  when  its  bases  are  in- 
scribed in  the  bases  of  the  cylinder. 
In  this  case,  the  cylinder  is  said  to 
be   circumscribed   about   the   prism. 

The    lateral    edges   of    the   inscribed 
prism    are    elements    of   the    surface   of 
the   circumscribing   cylinder. 


4.     A  prism  is    said    to    be    circum- 
scribed    about     a     cylinder-,     when     its 

bases  are  circumscribed  about  the  bases  of  the  cylinder. 
In  this  case,  the  cylinder  is  said  to  be  inscribed  in  the 
prism. 

The  straight  lines  which  join  the 
corresponding  points  of  contact  in  the 
upper  and  lower  bases,  are  common  to 
the  surface  of  the  cylinder  and  to  the 
lateral  faces  of  the  prism,  and  they 
are  the  only  lines  which  are  common. 
The  lateral  faces  of  the  prism  arc  said 
to  be  tangent  to  the  cylinder  along 
these  lines,  which  are  then  called  ele- 
ments  of  contact. 


5.  A  CoxB  is  a  volume  which  may  be  generated  by  a 
right-angled  triangle  revolving  about  one  of  the  sides  adja- 
cent to  the   right  angle,   as   an   axis. 


212 


GEOMETRY. 


Thus,  if  tlie  triangle  SAB,  right-angled  at  A,  be  tamed 
ahout  the  side  SA,  as  an  axis,  it  will  generate  the  cone 
S-CDBE. 

The  fixed  line  SA,  is  called  the 
axis  of  the  cone ;  the  curved  surface 
generated  by  the  hypothcnuse  SB^  is 
called  the  convex  surface  of  the  cone  ; 
the  circle  generated  by  tlie  side  AB^ 
is  called  the  base  of  the  cone ;  and 
the  iioiut  /S,  is  called  the  vertex  of 
the  cone  /  the  distance  from  the  vertex 
to  any  point  in  the  circumference  of  the 

base,  is  called  the  slant  height  of  the  cone ;  and  the  per- 
pendicular distance  from  the  vertex  to  the  plane  of  the  base, 
is   called   the   altitude   of  tlie   cone. 

The  line  SB,  Avhich  generates  the  convex  surface,  is,  in 
any  position,  called  an  element  of  the  surface  /  the  elements 
are  all  equal,  and  any  one  is  equal  to  the  slant  height  ;  the 
axis    is   equal  to   the  altitude. 

Any  line  of  the  generating  triangle  SAB,  as  GS, 
which  is  perpendicular  to  the  axis,  generates  a  circle  wh©se 
plane  is  perpendicular  to  the  axis  :  hence,  any  section  of  a 
cone  by  a  plane  perpendicular  to  the  axis,  is  a  circle.  Any 
section  SBC,  made  by  a  plane  through  the  axis,  is  an 
isosceles   triangle,   double   the   generating   triangle. 


6,  A  Truncated  Cone  is  that  portion  of  a  cone  included 
between   the   base    and    any    plane   which   cuts   the   cone. 

Wlien  the  cutting  plane  is  parallel  to  the  plane  of  the 
base,  the  truncated  cone  is  called  a  Frustum  of  a  Cone,  and 
the  intersection  of  the  cutting  plane  with  tl»e  cone  is  called 
the  upper  base  of  the  frustum  ;  tlie  base  of  the  cone  i? 
called   the   loicer   base  of  the   frustum. 


BOOK     VIII. 


213 


If  the  trapezoid  HGAB,  right-an- 
gled A  and  (r,  be  revolved  about 
AG^  as  an  axis,  it  will  generate  a  frus- 
tum of  a  cone,  whose  bases  are  EGDB 
and  FKH^  whose  altitude  is  AG^  and 
whose   slant   height   is    JBH. 


7.  Similar  Coxes  are  those  which  may  be  generated 
by  similar  right-angled  triangles  revolving  about  homologous 
sides. 

The  axes  of  similar  cones  are  proportional  to  the  radii 
of  their  bases  (B.  IV.,  D.  T)  ;  they  are  also  proportional  to 
any  other   homologous  lines    of  the   cones. 


8.  A  pyramid  is  said  to  be  in- 
scribed in  a  cone.)  when  its  base  is 
inscribed  in  the  base  of  the  cone,  and 
when  its  vertex  coincides  with  that  of 
the   cone. 

The  lateral  edges  of  the  inscribed 
pyramid  are  elements  of  the  surface  of 
the   circumscribing   cone. 


9.  A  pyramid  is  said  to  be  circumscribed  about  a  cone^ 
when  its  base  is  circumscribed  about  tlie  base  of  the  cone, 
and  when    its  vertex    coincides   with    that    of   the   cone. 

In  this  case,  the  cone  is  said  to  be  i?isc)'ibed  in  the 
pyramid. 

The  lateral  faces  of  the  circumscribing  pyramid  are  tan- 
gent to  the  surface  of  the  inscribed  cone,  along  Unes  which 
are  called  elements  of  contact. 


10.     A  frustum   of    a  pyramid   is   inscribed  in  a  frustum 


214 


GEOMETRY. 


of  a  C07ie^  wlien   its  bases   are  inscribed  in  the  bases   of  the 
frustum   of  the  cone. 

The  lateral  edges  of  the  inscribed  frustum  of  a  pyramid 
are  elements  of  the  surface  of  the  cii'cumscribing  frustum  of 
a  cone. 

11.  A   frustum    of    a    pyramid    is    circumscribed    about 
frustum   of  a   cone,   when    its    bases    are   cii'cumscribed    abouV . 
those   of  the   frustum   of  the   cone. 

Its  lateral  faces  are  tangent  to  the  surface  of  the  frustum 
of  the  cone,  along  Ihies  which   are  called  elements  of  contact. 

12.  A  Sphere  is  a  volume  bounded  by  a  surface,  every 
point  of  which  is  equally  distant  from  a  point  within  called 
the  centre. 

A  sphere  may  be  generated  by  a  semicircle  revolving 
about   its   diameter  as   an   axis. 

13.  A  Radius  of  a  sphere  is  a  straight  line  drawn  from 
the  centre  to  any  point  of  the  surface.  A  Diameter  is  any 
straiirht  line  drawn  through  the  centre  and  limited  at  both 
extremities  by   the   surface. 

All  the  radii  of  a  sphere  are  equal  :  the  diameters  are 
also   equal,   and  each  is  double  the  radius. 

14.  A  Spherical  Sector  is  a  volume  which  may  be  geis- 
erated  by  a  sector  of  a  circle  revolving  about  the  diameter 
passing  through   either  extremity   of  the   arc. 

The  surface  generated  by  the  arc  is  called  the  base  of* 
the  sector. 

15.  A  plane  is  Tangent  to  a  Sphere  when  it  touches 
it   in   a   single   point. 

16.  A  Zone  is  a  portion  of  the  surface  of  a  sj^here 
included    between    two    parallel    jjlanes.      The   bounding   lines 


BOOK     VIII. 


215 


ol  the    sections  are  called  bases  of  the  zone,  and  the  distance 
between   the   planes  is   called   the  altitude  of  the   zone. 

If  one  of  the  planes  is  tangent  to  the  sphere,  the  zone 
has  hut   one   base. 

17.  A  SrHEKicAL  Segment  is  a  portion  of  a  sphere  in- 
cluded between  two  parallel  planes.  The  sections  made  by 
the  planes  are  called  bases  of  the  segment,  and  the  distance 
between   them   is   called   the   altitude   of  the  segment. 

If  one  of  the  planes  is  tangent  to  the  sphere,  the  seg- 
ment has  but   one   base. 

The  Cylincee,  the  Cone,  and  the  Spheke,  are  sometunes 
called  The  Thkee  Round  Bodies. 


PKOPOSITIOIT     I. 


THEOKEM. 


The  convex  surface    of    a    cylinder  is    equal    to   the  cireunir 
ference  of  its  base  midtiplied  by  the  altitude. 

Let  ABD  be  the  base  of  a  cylinder  whose  altitude  is 
U :  then  will  its  convex  surface  be  equal  to  the  circunv 
ference   of  its   base   multij)lied   by   the   altitude. 

For,  inscribe  within  the  cylinder  a 
prism  whose  base  is  a  regular  polygon. 
The  convex  surface  of  this  prism  will 
be  equal  to  the  perimeter  of  its  base 
multiplied  by  its  altitude  (B.  VU.,  P.  I,), 
whatever  may  be  the  number  of  sides 
of  its  base.  But,  when  the  number  of 
sides  is  infinite  (B.  V.,  P.  X.  Sch.),  the 
convex  surface  of  the  prism  coincides  with 
that    of  the    cylinder,    the    permieter    of  b 


i» 


C 


J 


I) 


216 


GEOMErHY. 


the  base  of  the  iirisra  coincides  with  the  circumference  of 
the  base  of  the  cylinder,  and  the  altitude  of  the  prism  is 
the  same  as  that  of  the  cyUnder  :  hence,  the  convex  surf-ice 
of  the  cyhnder  is  equal  to  tlie  circumference  of  its  base 
multiplied  by  the   altitude  ;    whicJi  was   to  he  proved. 

Cor.    The   convex   surfaces    of   cyUnders  having   equal   alti- 
tudes  are   to   each   other  as  the  circumference^  of  their  bases. 


PROPOSITION     II. 


THEOREM. 


Tlie    volume    of   a    cylinder  is   equal  to    the   product  of  its 

base  and  altitude. 

Let  ABB  be  the  base  of  a  cylinder  Avhose  altitude  is 
R ;  then  will  its  volume  be  equal  to  the  product  of  its 
base   and   altitude. 

For,  inscribe  within  it  a  prism  whose 
base  is  a  regular  polygon.  The  volume 
of  this  prism  is  equal  to  the  product 
of  its  base  and  altitude  (B.  VII.,  P. 
XrV.),  whatever  may  be  the  number  of 
sides  of  its  base.  But,  when  the  num- 
ber of  sides  is  infinite,  the  prism  coin- 
cides with  the  cyUnder,  the  base  of  the 
prism  with  the  base  of  the  cylinder,  and 
the    altitude    of    the    prism   is   the   same 

ai  that  of  the  cylinder  :  hence,  the  volume  of  the  cylinder 
is  equal  to  the  product  of  its  base  and  altitude  ;  which  toas 
to  be  proved. 

Cor.  1.  Cylinders  are  to  each  other  as  the  products  of 
their  bases  and  altitudes ;  cylinders  having  equal  bases  are 
to  each  other  as  their  altitudes  ;  cylinders  having  equal  alii, 
tudes   are   to   each    other   as   their  bases. 


BOOK     VIII. 


217 


Cor.  2.  Similar  cylinders  are  to  each  other  as  the  (iubes 
of  their  altitudes,  or  as  the  cubes  of  the  radii  of  their 
bases. 

For,  the  bases  are  as  the  squares  of  their  radii  (B.  V., 
P.  Xin.),  and  the  cylinders  being  similar,  these  radii  arc  to 
each  other  as  their  altitudes  (D.  2)  :  hence,  the  bases  are 
s  the  squares  of  the  altitudes  ;  therefore,  the  bases  multiplied 
by  the  altitudes,  or  the  cylinders  themselves,  are  as  the 
cubes   of  the   altitudes. 


PEOrOSITIO^     III.        THEOREM. 

The  convex  surface   of  a  cone  is  equal  to  the  circumference 
of  its  base  multiplied  by,  half  the  slant  height. 

Let  S-ACD  be  a  cone  whose  base  is  ACB^  and  whose 
slant  height  is  SA  :  then  will  its  convex  surface  be  equal 
to  the  circumference  of  its  base  multiplied  by  half  the  slant 
height. 

For,  inscribe  within  it  a  right  pyramid. 
The  convex  surface  of  this  pyramid  is 
equal  to  the  perimeter  of  its  base  mul- 
tiplied by  half  the  slant  height  (B.  VII., 
P.  rV.),  whatever  may  be  the  number 
of  sides  of  its  base.  But  when  the  num- 
ber of  sides  of  the  base  is  infinite,  the 
convex  surface  coincides  with  that  of  the 
cone,  the  perimeter  of  the  base  of  the  pyramid  coincides  with 
the  circumference  of  the  base  of  the  cone,  and  the  slant  height 
of  the  pyramid  is  equal  to  the  slant  height  of  the  cone : 
hence,  the  convex  surface  of  the  cone  is  equal  to  the  cir- 
cumference of  its  base  multiplied  by  half  the  slant  height ; 
lohich  was  to  be  proved. 


218 


GEOMETRY. 


PROPOSITION     IV.        THEOREM. 

The  convex  surface  of  a  frustum  of  a  cone  is  equal  to 
half  the  sum  of  the  circumferences  of  its  two  bases 
multiplied  by  the  slant  height. 

Let  BIA-D  be  a  frustum  of  a  cone,  BIA  and  EGD 
its  two  bases,  and  EB  its  slant  height :  then  is  its  convex 
surface  equal  to  half  the  sum  of  the  circumferences  of  its 
two   bases   multiplied  by  its  slant  height. 

For,  inscribe  within  it  the  frustum 
of  a  right  pyramid.  The  convex  sur- 
face of  this  frustum  is  equal  to  half 
the  sum  of  the  perimeters  of  its  bases, 
multiplied  by  the  slant  height  (B.  VII., 
P.  IV.,  C),  whatever  may  be  the 
number  of  its  lateral  faces.  But  when 
the   number   of  these   faces  is  infinite, 

the  convex  surface  of  the  frustum  of  the  pyramid  coincides 
with  that  of  the  cone,  the  perimeters  of  its  bases  coincide 
with  the  circumferences  of  the  bases  of  the  frustum  of  the 
cone,  and  its  slant  height  is  equal  to  that  of  the  cone  : 
hence,  the  convex  surface  of  the  frustum  of  a  cone  is  equal 
to  half  the  sum  of  the  circmnferences  of  its  bases  multiplied 
by  the   slant   height ;    which  was   to   be  2^oved. 

Scholium.  From  the  extremities  A  and  Z>,  and  from 
tho  middle  point  ^,  of  a  hne  AD^  let  the  lines  AO,  DC^ 
and  IK.,  be  drawn  perpendicular  to  the  axis  0C-.  then  Avill 
IK  be  equal  to  half  the  sum  of  AO  and  I) C.  For, 
draw  Bd  and  li^  perpendicular  to  ^  0 :  then,  because  Al 
is  equal  to  IB^  we  shaU  have  Ai  equal  to  id  (B.  IV.,  P. 
XV.),    and    consequently    to    Is  ;    that    is,    A  0    exceeds    IK 


BOOK     VIII.  219 

as  much  as  IIC  exceeds  D  G :  hence,  IIK  is  equal  to  the 
half  sum   of   -4  0    and  DG. 

NoM',  if  the  lino  AD  be  revolved  about  0(7,  as  an 
axis,  it  will  generate  the  surface  of  a  frustum  of  a  cone 
whose  elant  height  is  AD ;  the  point  I  will  generate  a 
^ii'cumference  which  is  equal  to  half  the  sum  of  the  cu-cum- 
erences  generated  by  A  and  D :  hence,  if  a  straight  line 
^e  revolved  about  another  straight  line,  it  will  generate  a 
surface  whose  measure  is  equal  to  the  product  of  the  gene- 
rating  line  and  the  circumference  generated  by  its  middle 
point. 

This  proposition  holds  true  when  the  line  AD  meets 
00,    and    also  when   AD    is    parallel    to     OG. 

PROPOSITION      V.        THEOREM. 

T^ie  volume    of   a    cotie    is    equal  to  its    base  m^ultiplied  by 

one-third  of  its  altitude. 

Let  ABDE  be  the  base  of  a  cone  whose  vertex  is  /S, 
and  whose  altitude  is  So  :  then  will  its  volume  be  equal  to 
the  base  multiplied  by   one-third   of  the   altitude. 

For,    inscribe    in    the    cone    a   right 
pyramid.      The  volume  of  this  pyramid  A 

is   equal   to   its  base   multiplied   by  one-  / //i  V\ 

thii-d  of  its  altitude  (B.  VH.,  P.  XVIL),  / / 1\  \\ 

whatever    may    be    the    number    of   its  y^'^'^  f l-^«AA 

lateral    faces.      But,   when    the    number         ^^ J  ]^    \  A 

of  lateral   faces   is   infinite,    the   pyramid  \\      /       ^W 

coincides  with    the    cone,    the    base    of  jg 

the   pyramid   coincides  with  that   of  the 

cone,   and   their  altitudes   are   equal  :    hence,  the   volume   of  a 

cone    is    equal    to    the    base    multiplied    by   one-third   of    the 

altitude  ;    which  was  to   be  proved. 


220  GEOMETRY. 

Cor.  1.  A  cone  is  equal  to  one-third  of  a  cylinder  hav- 
ing  an   equal  base   and  an   equal   altitude. 

Cor.  2.  Cones  are  to  each  other  as  the  products  of 
their  bases  and  altitudes.  Cones  having  equal  bases  are  to 
each  other  as  their  altitudes.  Cones  having  equal  altitudes 
are   to   each   other  as  their  bases. 


PROPOSITIOIT      VI.        THEOEEM. 

The  'dolume  of    a    frustum   of  a  C07ie  is  equal  to   the   sum 
of  the    volumes    of    three    cones^    having  for    a    co7nmon 
altitude    the    altitude    of   the   frustum^   and  for  bases   the 
lower  base  of   the   frustum^   the  upper  base    of   the   frus 
tum^   and  a  inean  proportional  between  the  bases. 

Let  BIA  be  the  lower  base  of  a  frustum  of  a  cone, 
EGB  its  upper  base,  and  00  its  altitude  :  then  will  its 
volume  be  equal  to  the  sum  of  three  cones  whose  common 
altitude  is  0(7,  and  whose  bases  are  the  lower  base,  the 
upper   base,    and   a   mean   proportional    between   them. 

For,  inscribe  a  frustum  of  a  right 
pyramid  in  the  given  frustum.  The 
volume  of  this  frustum  is  equal  to 
the  sum  of  the  volumes  of  three 
pyramids    whose    common    altitude    is 

that   of  the  frustum,  and  whose   bases       ^^ 

re  the  lower  base,  the  upper  base, 
and  a  mean  proportional  between  the 
two     (B.  VII.,    P.  XVm.),    whatever 

may  be  the  number  of  lateral  faces.  But  when  the  numbei 
of  faces  is  infinite,  the  frustum  of  the  pyramid  coincides 
with  the  frustum  of  the  cone,  its  bases  with  the  bases  of 
the  cone,  the  three  pyramids  become  cones,  and  their  altitudes 


BOOK     VIII. 


221 


are  equal  to  that  of  tlie  frustum  ;  hence,  the  volume  of  the 
frustum  of  a  cone  is  equal  to  the  sura  of  the  volumes  of 
three  cones  whose  common  altitude  is  that  of  the  frustum, 
and  whose  bases  are  the  lower  base  of  the  frustum,  the 
upper  base  of  the  frustum,  and  a  moan  proportional  between 
ill  em  ;    which   loas   to   he  i^oved. 


PROPOSIXrON      VII. 


TnEOREM. 


Any  section   of  a   sphere  made   hy   a  plane,   is  a   circle. 

Let  C  be  tlie  centre  of  a  sphere,  CA  one  of  its 
radii,  and  A^FB  any  section  made  by  a  plane :  then  Tvill 
this  section   be   a   circle. 

For,  draw  a  radius  CO  perpen- 
dicular to  the  cutting  plane,  and  let 
it  pierce  the  2')lane  of  the  section  at 
0.  DraAv  radii  of  the  sphere  to  any 
two  points  J/,  3I\  of  tlie  curve  which 
bounds  the  section,  and  join  these 
points  with  0  :  then,  because  the  radii 
CM,       CM'       are     equal,     the     points 

J/,    il/',    will   be  equally  distant   from    0   (B.  VT.,  P.  V.,  C.)  ; 
hence,   the   section   is   a   circle  ;    xchich  was  to  he  pyroved. 

Cor.  1.  When  the  cutting  plane  passes  through  the  centre 
of  the  sphere,  the  radius  of  the  section  is  equal  to  that  of 
the  sphere ;  when  the  cutting  plane  does  not  p.iss  through 
the  centre  of  the  sphere,  the  radius  of  the  section  will  be 
loss  than   that   of  the   sphere. 

A  section  whose  plane  passes  through  the  centre  of  the 
sphere,  is  called  a  great  circle  of  the  sphere.  A  section 
whose  plane   does  not   pass  through  the   centre  of  the    sphere, 


222  GEOMETRY. 

is    called   a   small  circle  of  the   sphere.      All   great   circles   of 
the   same,   or   of  equal   spheres,   are   equal. 

Cor.  2.  Any  great  circle  divides  the  sphere,  and  also 
the  surface  of  the  sphere,  into  equal  parts.  For,  the  parts 
may  be  so  placed  as  to  coincide,  other^vise  there  would  bo 
some  points  of  the  surface  unequally  distant  from  the  centre, 
which   is   impossible. 

Cor.  3.  The  centre  of  '  a  sphere,  and  the  centre  of  any 
small  circle  of  that  sphere,  are  in  a  straight  line  perpen- 
dicular  to   the   plane   of  the    circle. 

Cor.  4.  The  square  of  the  radius  of  any  small  circle  is 
equal  to  the  square  of  the  radius  of  the  sphere  diminished 
by  the  square  of  the  distance  from  the  centre  of  the  sphere 
to  the  plane  of  the  circle  (B.  IV.,  P.  XL,  C.  1)  :  hence, 
circles  which  are  equally  distant  from  the  centre,  are  equal  ; 
and  of  two  circles  which  are  unequally  distant  from  the 
centre,  that  one  is  the  less  whose  plane  is  at  the  greater 
distance   from   the   centre. 

Cor.  5.  The  circumference  of  a  great  circle  may  always 
be  made  to  pass  through  any  two  points  on  the  surface  of 
a  sphere.  For,  a  plane  can  always  be  passed  through  these 
points  and  the  centre  of  the  sphere  (B.  VI.,  P.  II.),  and  its 
section  will  be  a  great  circle.  If  the  two  points  are  the 
extremities  of  a  diameter,  an  infinite  number  of  planes  caci 
be  ]>assed  through  them  and  the  centre  of  the  sphere  (B.  VT^ 
P.  I.,  S.)  ;  in  this  case,  an  infinite  number  of  great  circles 
can   be   made   to   pass   through   the   two  points. 

Cor.  6.  The  bases  of  a  zone  are  the  circumferences  of 
circles  (D.  16),  and  the  bases  of  a  segment  of  a  sphere  are 
circles. 


BOOK     VIII. 


223 


PROPOSITION     VIII. 


TIIEORE]\r. 


Any  plane  perpendicular  to  a  radius  of  a  spJiere  at  its  outer 
extremity,  is  tangent  to  the  spliere  at  that  point. 

Let  G  be  the  centre  of  a  sphere,  CA  any  radius,  and 
FAG  a  plane  perpendicular  to  GA  at  A  :  then  will  the 
plane    FA  G    be   tangent   to   the   sphere   at    A. 

For,  from  any  other  point  of  the 
plane,  as  J/,  draw  the  line  3IG : 
then  because  GA  is  a  i^erpendicular 
to  the  plane,  and  G3T  an  oblique 
line,  GM  will  be  greater  than  GA 
(B.  YI.,  P.  V.)  :  hence,  the  point  31 
lies  without  the  sphere.  The  plane 
FAG^     therefore,   touches    the    sphere 

at    A^     and    consequently    is    tangent    to    it    at    that    point  , 
which  was   to   be  proved. 

Scholium.  It  may  be  shown,  by  a  course  of  reasoning 
analogous  to  that  employed  in  Book  m.,  Propositions  XI,, 
Xn.,  Xin.,  and  XIV.,  that  two  spheres  may  have  any  one 
of  six   positions   with   respect   to    each    other,   viz,  : 

1°.  When  the  distance  between  their  centres  is  greater  than 
the    sum    of    their  radii,   they  are  external.,  one  to  the  other  : 

2°.  "WTien  the  distance  is  equal  to  the  sum  of  their 
radii,   they  are  tangent.,  externally  : 

3".  When  this  distance  is  less  than  the  sum,  and  greater 
than   the   difference   of  their  radii,   they  iyitersect  each  other : 

4°.  When  this  distance  is  equal  to  the  difference  of  their 
radii,   they   are   tangent  internally : 

5°.  When  this  distance  is  less  than  the  difference  of  their 
radii,   one  is  wholly  within  the  other  : 

6*.  When  this  distance  is  equal  to  zero,  they  have  a 
common  centre^   or,  are   concentric. 


224 


GEOMETRY. 


DEFINITIONS. 

1°.  If  a  semi-circumference  be  divided  into  equal  arcs,  the 
chords  of  these  arcs  form  half  of  the  perimeter  of  a  regular 
inscribed  polygon  ;  this  half  j)erimeter  is  called  a  regular 
semi-perimeter.  The  figure  bounded  by  the  regular  semi- 
perimeter  and  the  diameter  of  the  semi-circumference  is  called 
a  regular  semi-polygon.  The  diameter  itself  is  called  the 
a.xis  of  the   semi-polygon. 

2°.  If  lines  be  drawn  from  the  extremi- 
ties of  any  side,  and  perpendicular  to  the 
axis,  the  intercepted  portion  of  the  axis  is 
called  the  projection  of  that   side. 

The  broken  Hue  ABGDGP  is  a  regu- 
lar semi-perimeter  ;  the  figure  bounded  by 
it  and  the  diameter  AP^  is  a  regular 
semi-polygon,  AP  is  its  axis,  MIT  is  the 
projection  of  the  side  j??(7,  and  the  axis, 
AP^    is  tlie   i^rojection   of  the   entire   semi-perimeter. 


PEOPOSITION"     IX. 


LEJIMA. 


If  a    regular    semi-polygon    he    revolved   about    its  axis,   the 

surface  generated  by  the    semi-perimeter  will    be  equcd    to 

the   axis  r)%ultiplied    by   the  circumference  of  the  inscribed 
circle. 

Let  ABCDEF  be  a  regular  semi-polygon,  AF  its  axis, 
and  ON  its  apothem  :  then  will  the  surfoce  generated  by 
the   regular   semi-perimeter  bo   equal  to    AF  x  circ.  OJV. 

From  the  extremities  of  any  side,  as  DF,  draw  J)I 
and  FH  perpendicular  to  AF ;  draw  also  Nlff  perpen- 
dicular to  AF,  and  FIT  perpendicular  to  DI.  Now,  the 
surface    generated    by     FD     is    equal    to     DF  x  circ.  NM 


BOOK     VIII. 


225 


(P.  TV.,  S.).      But,   because   the   trianglea    UBK  and     OJVM 
are   similar   (B.  IV.,  P.  XXI.),   we   have, 

I)£J  :   EK    or  JR  :  :   ON   :   JSfM  :  :  circ.ON  :   cire.JVM] 


whence, 

DB  X  cire.  JVM 


IJT  X  circOJSr  ; 


that  is,  the  surface  generated  by  any  side 
is  equal  to  the  projection  of  that  side 
multiplied  by  the  circumference  of  the  in- 
scribed circle  :  hence,  the  surface  gene- 
rated by  the  entire  semi-perimeter  is  equal 
to  the  sum  of  the  projections  of  its  sides, 
or  the  axis,  multiplied  by  the  circumfer- 
ence  of  the  inscribed   circle  ;    xohich  was   to   he  proved. 

Cor.  The  surface  generated  by  any  portion  of  the  perim- 
eter, as  CBE,  is  equal  to  its  projection  PR,  multiplied 
by  the   circumference   of  the   inscribed   circle. 

PEOPOSITIOK      X.        THEOEEM. 

The  surface  of  a  sphere  is  equal  to  its   diameter  midtipUed 
hy  the  circumference  of  a  great  circle. 

Let  ABODE  be  a  semi-circumference, 
0  its  centre,  and  AE  its  diameter :  then 
will  the  surface  of  the  sphere  generated 
by  revolving  the  semi-circumference  about 
AE^    be  equal   to    AE  x  circ.  OE. 

For,  the  semi-circumference  may  be  re- 
garded as  a  regular  semi-perimeter  with  an 
infinite  number  of  sides,  whose  axis  is  AE^ 
and  the  radius  of  whose  inscribed  circle 
is    OE  :    hence  (P.  IX.),  the  surface  generated  by  it  is  equal 

to    AE  X  circ.  OE;    which  was  to  be  proved. 

15 


^A 


22G 


GEOMETRY. 


Cor.  1.    The    circumference   of  a    great   circle    is    equal   to 
2  T  OE    (B.  v.,   P.  XVI.)  :    hence,    the    area    of    the    surface 
of    the   sphere   is   equal  to     2  0E  X   2'kOE,      or   to    AitOE' 
that  is,  the  area  of  the  surface  of  a  sphere  is  equal  to  fouv 
great  circles. 

Cor.  2.  The  surface  generated  by  any 
arc  of  the  semicu'cle,  as  i?C,  will  be  a 
zone,  whose  altitude  is  equal  to  the  pro- 
jection of  that  arc  on  the  diameter.  But, 
the  arc  i5(7  is  a  portion  of  a  semi- 
perimeter  having  an  infinite  number  of 
sides,  and  the  radius  of  whoso  inscribed 
circle  is  equal  to  that  of  the  sphere : 
hence  (P.  IX.,  C),  the  surface  of  a  zone 
is  equal  to  its  altitude  multiplied  by  the  circumference  of  a 
great   circle   of  the   sjihere. 

Cor.  3.     Zones,   on  the   same   sphere,   or   on   equal  spheres, 
are  to   each   other   as  their   altitudes. 


PROPOSITION     XI.         LEMMA. 

If  a  triangle  and  a  rectangle  having  the  same  base  and 
equal  altitudes^  he  revolved  about  the  common  base^  tlie 
volume  generated  by  the  triangle  will  be  one-third  of  that 
generated  by  the  rectangle. 

Let  ABC  be  a  triangle,  and  EFBC  a  rectangle,  having 
the  same  base  BC^  and  an  equal  altitude  AB^  and  let 
tliera  both  be  revolved  about  BC:  then  will  the  volume 
generated  by  ABC  be  one-third  of  that  generated  by 
EFBC. 

For,  the  cone  generated  by  the  right-angled  triangle 
ADBy     is   equal    to    one-third    of    the   cylinder  generated    by 


BOOK    VIII. 


227 


the  rectangle  ADBF  (P.  V.,  C.  1),  and  the  cone  generated 
by  tlie  triangle  ADC,  is  equal  to  one-third  of  the  cylinder 
generated  by  the  rectangle  ABCE. 

When  AD  falls  within  the  triangle,  the 
sum  of  the  cones  generated  by  ADB  and 
ADC,  is  equal  to  the  yolume  generated  by 
the  triangle  ABC;  and  the  sura  of  the 
cylinders  generated  by  ADBF  ani^  ADCE, 
is  equal  to  the  volume  generated  by  the 
rectangle  EFBC. 

When  AD  falls  without  the  triangle,  the  difference  of  the  cones 
generated  hj  ADB  and  ADC,  is  equal  to  the  volume  generated  by 
ABC;  and  the  difference  of  the  cylinders 
generated  by  ADBF  and  ADCE,  is  equal 
to  the  volume  generated  by  EFBC:  hence, 
in  either  case,  the  volume  generated  by 
the  triangle  ABC,  is  equal  to  one-third  of 
the  yolume  generated  by  the  rectangle 
EFBC;   loMcli  was  to  he  proved. 

Cor.  The  volume  of  the  cylinder  generated  by  EFBC,  is 
equal  to  the  product  of  its  base  and  altitude,  or  to  -r  AD^  X  BC: 
hence,  the  volume  generated  by  the  triangle  ABC,  is  equal  to 
i'^AD'^xBC. 


PROPOSITIOX    XTI.      LEMMA. 

If  an  isosceles  triangle  ie  revolved  about  a  straiglit  line 
passing  through  its  vertex,  the  volume  generated  will  ie 
equal  to  the  surface  generated  hy  the  base  multiplied  by 
one-third  of  the  altitiide. 

Let  CAB  be  an  isosceles  triangle,  C  its  vertex,  AB  ita 
base,  CI  its  altitude,  and  let  it  be  revolved  about  the  line  CB, 
as  an  axis:  then  will  the  volume  generated  be  equal  to  su^f 
AB  X  i  CI.    There   may  be  three   cases: 


228 


GEOMETRY. 


1°.  Suppose  the  base,  when  produced,  to  meet  the  axis  at 
D\  draw  AM,  IK,  and  BN, 
perpendicular  to  CD,  and  BO 
parallel  to  DC.  Now,  the 
volume  generated  by  CAB  is 
equal  to  the  difference  of  the 
volumes  generated  by  CAD  and 
CBD;  hence  (P.  XL,  C), 

vol.  CAB=i':rAM'x  CD-i'rBW'x  CD=^ir{lM^-BN')  X  CD. 

But,  AM^  -  BN^  is  equal  to  {AM  +  BN)  {AM  -  BN), 
(B.  IV.,  P.  X.)  ;  and  because  AM  +  BN  is  equal  to  2IK 
(P.  IV.,  S.),   and  A3I  -  BN  to  AG,  we  have, 

vol.   CAB  =  \'K  IKx  AO  X  CD. 

But,  the  right-angled  triangles  AOB  and  CDI  are  similar 
(B.   IV.,   P.  XVIII. ;  hence, 

AO    '.    AB    '.'.    CI    :    CD;    or,    AO  X   CD  =  AB  X   CI. 

Substituting,  and  clianging  the  order  of  the  factors,  we  have, 

vol.  GAB  =  AB  X  2  '!f  IK  X  y  CI. 

But,   AB  X  2 'jf  lie  =  the  surface  generated  by  AB ;    hence, 

vol.   CAB  =  surf.  AB  X  i  CI 


2°.    Suppose  the  axis  to  coincide  with  one  of  the  equal  sides. 

Draw  C/ perpendicular  to  ^-S,and  AM 
and  IK,  perpendicular  to  CB.    Then, 

vol   CAB  =  ^  *  AM^  xGB  =  \'g  AMx 
AMX  CB. 


But,   since  A  MB  and    CIB    are   similar, 
JM     :    AB    :  :   CI 


c  M     s:    B 

CB',    whence  AM  X  CB  =  AB  X  OL 
Also,  AM  =  2  IK',    hence,  by  substitution,   we  have, 

vol  CAB  =  ABX  2if  IKx\CI=  surf.  AB  X  i  CI. 


BOOK    VIII. 


229 


3°.     Suppose  the  base  to  be  parallel   to   the   axis. 

Draw    AM    and    BH    perpendicular    to     the    axis.       The 
volume    generated    by    CAB,    is    equal  A  I  B 

to  the  cylinder  generated  by  the  rectan- 
gle ABNM,  diminished  by  the  sum  of 
the  cones  generated  by  the  triangles 
CAM  and  BGN;    hence, 


vol.   CAB  =  It  Cr  X  AB  -  ^  •!(  Cr  X  AI  -\-K  CI   X  IB. 

But    the    sum    of    AI   and  IB  is  equal  to  AB :    hence,  we 
have,   by  reducing,   and  changing  the  order  of  the  factors, 

vol.  CAB  =  AB  X^^  CI  X\  CL 

But  AB  X  '^  <!(  CI  is  equal  to  the  surface  generated  hj  AB; 
consequently, 

vol.   CAB  =  surf.  AB  X  ^  CI; 

hence,   in  all  cases,  the  volume  generated  by    CAB  is  equal 
to  surf.  AB  X  ^  CI;   which  was  to  ie  jjroved. 


PROPOSITION   XIII.      LEMMA. 

If  a  regular  semi-polygon  he  revolved  about  its  axis,  the  volume 
generated  will  he  equal  to  the  surface  generated  hj  the  semi- 
perimeter  multiplied  hy  one-third  of  the  apothem. 

Let  FBDG  be  a  regular  semi-poly- 
gon, FG  its  axis,  01  its  apothem,  and 
let  the  semi-polygon  be  revolved  about 
FG  :  then  will  the  volume  generated 
be   equal   to     surf.  FDBG  x  \0L 

For,  draw  lines  from  the  vertices  to 
the  centre  0.  These  lines  will  divide 
the  semi-polygon  into  isosceles  triangles 
whose  bases  are  sides  of  the  semi-polygon, 
and   whose   altitudes   are   equal  to     OL 


230 


GEOMETRY. 


Now,  the  sum  of  the  volumes  generated  by  these  trian- 
gles is  equal  to  the  volume  generated  by  the  semi-polygon. 
But,  the  volume  generated  by  any  triangle,  as  OAB,  is 
equal  to  surf.  AB  X  \0I  (P.  XII.)  :  hence,  the  volume 
generated  by  the  semi-polygon  is  equal  to  surf.  FBD G  x  \0I ', 
ohich  was  to   be  jp^'ouec?. 

Cor.     The   volume    generated    by   a    portion   of    the   semi  ■ 
polygon,    OABCy    limited    by   radii    0(7,      OA^     is  equal  to 
»icrf  ABC  X  \0I. 


PEOPOSITION      XIV.        THEOREM. 

The  volume  of  a    sphere    is    equal  to  its  surface  multiplied 
by  one-third  of  its  radius. 

Let  ACE  be  a  semicircle,  AE  its 
diameter,  0  its  centre,  and  let  the  semi- 
cu-cle  be  revolved  about  AE\  then  will 
the  volume  generated  be  equal  to  the 
surface  generated  by  the  semi-circumfer- 
ence multiplied  by  one-third  of  the  radius 
OA. 

For,  the  semicircle  may  be  regarded 
as  a  regular  semi-polygon  having  an  infi- 
nite number  of  sides,  whose  semi-perimeter 
coincides  with  the  semi-circumference,  and  whose  apothem  is 
equal  to  the  radius :  hence  (P.  Xm.),  the  volume  gene- 
rated by  the  semicircle  is  equal  to  the  surface  generated  by 
the  semi-circumference  multiplied  by  one-third  of  the  radius  ; 
which  was   to  be  proved. 

Cor.  1.    Any   portion    of   the   semicircle,  as   OBC,  bounded 
by   two   radii,   will    generate    a   volume    equal   to   the    surface 


BOOK     VIII. 


231 


generated  by  the  arc  BG  multiplied  by  one-third  of  tlie 
radius  (P.  XIII.,  C).  But  this  portion  of  the  semicircle  ia 
a  circular  sector,  the  volume  which  it  generates  is  a  spheri- 
cal sector,  and  the  surface  generated  by  the  arc  is  a  zone  : 
hence,  the  volume  of  a  spherical ,  sector  is  equal  to  the  zone 
w/iich   forms  its   base  multiplied  by  one-third  of  the  radiu» 

Cor.  2.  If  we  denote  the  volume  of  a  sphere  by  F", 
and  its  radius  by  i?,  the  area  of  the  surface  will  be  equal 
to  4wi22  (P.  X.,  C.  1),  and  the  volume  of  the  sphere  will  be 
equal    to    '^ 'r( U"^  x  ^ R  \     consequently,  we   have, 

Again,  if  we  denote  the  diameter  of  the  sphere  by  Z>,  we 
shall  have  JR,  equal  to  ^Z>,  and  R^  equal  to  \D^^  and 
consequently, 

hence,  the  volumes  of  spheres  are  to  each  other  as  the  cubes 
of  their  radii^    or  as   the  cubes  of  their  diameters. 

Scholium.  If  the  figure  EBDF,  formed 
by  drawing  lines  from  the  extremities  of  the 
arc  BD  perpendicular  to  CA,  be  revolved 
about  CA,  as  an  axis,  it  will  generate  a  seg- 
ment of  a  sphere  whose  volume  may  be  found 
by  adding  to  the  spherical  sector  generated  by 
CDB,  the  cone  generated  by  CBE,  and  sub- 
tracting from  their  sum  the  cone  generated 
iby  CDF.  If  the  arc  BD  is  so  taken  that  the 
points  E  and  F  fall  on  opposite  sides  of  the  centre  C,  the 
latter  cone  must  be  added,  instead  of  subtracted:  zone  BD 
-^•g  CD  y.  EF\    hence, 

segment  EBDF  =  ^  *  (2  'W  X  EF  +  BE''  X  CE  ^  DT  X  CF). 


232 


GEOMETRY. 


PEOPOSITION"     XV.        THEOKEM 

The  surface  of  a  sphere  is  to  the  entire  surface  of  the 
circumscribed  cylinder,  including  its  bases,  as  2  is  to  3  ; 
and  the  volumes  are  to  'each  other  in  the  same  ratio. 

Let  PMQ  be  a  semicircle,  and  PADQ  a  rectangle, 
whose  sides  PA  and  QD  are  tangent  to  the  semicircle  at 
P  and  Q,  and  whose  side  AD,  is  tangent  to  the  semi- 
circle at  3f.  If  the  semicircle  and  the  rectangle  be  revolved 
about  PQ,  as  an  axis,  the  former  will  generate  a  sphere, 
and  the   latter   a   circumscribed   cylinder. 

1°.  The  surface  of  the  S2:)here  is  to  the  entii'e  surface  of 
the   cylinder,   as    2    is  to    3. 

For,  the  surface  of  the  sphere  is 
equal  to  four  great  cii'cles  (P.  X.,  C.  1), 
the  convex  surface  of  the  cylinder  is 
equal  to  the  circumference  of  its  base 
multiplied  by  its  altitude  (P.  I.)  ; 
that  is,  it  is  equal  to  the  circumfer- 
ence of  a  great  circle  multiplied  by 
its  diameter,  or  to  four  great  circles 
(B.  v.,    P.  XV.)  ;    adding   to   this  the 

two  bases,  each  of  which  is  equal  to  a  great  cii'cle,  we  have 
the  entire  surface  of  the  cylinder  equal  to  six  great  cu'cles : 
hence,  the  surface  of  the  sphere  is  to  the  entire  surface  of 
Le  circumscribed  cylinder,  as  4  is  to  6,  or  as  2  is  to  3  ; 
which  was   to  be  proved. 

2".  The  volume  of  the  sphere  is  to  the  volume  of  the 
cylinder   as    2    is   to    3. 

For,  the  volume  of  the  sphere  is  equal  to  f  ^i^^  (P.  XH^., 
C.  2)  ;  the  volume  of  the  cylinder  is  equal  to  its  base 
multiplied    by  its    altitude   (P.  II.)  ;    that    is,   it   is    equal    to 


BOOK     VIII.  233 

kR^  X  2Ji,  or  to  f  irH^  :  hence,  tlie  volume  of  the  sphere 
Is  to  that  of  the  cylmder  as  4  is  to  C,  or  as  2  is  to  3  • 
lohich  was   to  be  2^^oved. 

Cor.  The  surface  of  a  sphere  is  to  the  entire  surface  of 
a  circumscribed  cyHnder,  as  the  volume  of  the  sphere  is  to 
volume   of  the   cylinder. 

Scholium.  Any  polyedron  which  is  circumscribed  about  a 
sphere,  that  is,  whose  faces  are  all  tangent  to  the  sphere, 
may  be  regarded  as  made  up  of  pyramids,  whose  bases  are 
the  faces  of  the  polyedron,  whose  common  vertex  is  at  the 
centre  of  the  sphere,  and  each  of  whose  altitudes  is  equal 
to  the  radius  of  the  sphere.  But,  the  volume  of  any  one 
of  these  pyramids  is  equal  to  its  base  multii^lied  by  one- 
third  of  its  altitude :  hence,  the  volume  of  a  circumscribed 
polyedron  is  equal  to  its  surface  multij^Ued  by  one-third  of 
the   radius   of  the   inscribed   sphere. 

Now,  because  the  volume  of  the  sphere  is  also  equal  to 
its  surface  multijjlied  by  one-third  of  its  radius,  it  follows 
that  the  volume  of  a  sphere  is  to  the  volume  of  any  cir- 
cumscribed polyedron,  as  the  surface  of  the  sphere  is  to  the 
surface   of  the   polyedron. 

Polyedrons  circumscribed  about  the  same,  or  about  equal 
spheres,   are   proportional  to   their   surfaces. 


GENERAL      rOEMULAS. 

If  we  denote  the  convex  surface  of  a  cylinder  by  S^  its 
volume  by  V]  the  radius  of  its  base  by  H,  and  its  alti« 
tude  by    jff",     we  have   (P.  I.,  11.), 

S  =  2i(R  X  H (1.) 

V  =    '^R  X  H (2.) 


231  GEOMETRY. 

If  we  denote  the  convex  surface  of  a  cone  by  S,  its 
volume  by  V]  the  radius  of  its  base  by  M,  its  altitude  by  H^ 
and  its  slant  height  by  JF,  we  have  (P.  III.,  V.), 

S  =   -rrJix  II' (3.) 

F=  *i2'x  lir (4.) 

If  we  denote  the  convex  surface  of  a  frustum  of  a  cone 
by  /S,  its  volume  by  V,  the  radius  of  its  lower  base  by  ^, 
the  radius  of  its  upper  base  by  H',  its  altitude  by  My  and  its 
slant  height  by  M\  we  have  (P.  IV.,  VI.), 

JS  =  '^{H  +  FJ)  X  IT' (5.) 

V=  i'!riIi'  +  B"  +  Mx  H')  X  JI .    .     .     (6.) 

If  we  denote  the  surface  of  a  sphere  by  S^  its  volume 
by  Y,  its  radius  by  i?,  and  its  diameter  by  Z>,  we  have 
(P.  X.,  C.  1,    XIV.,  C.  2,  XIV.,  C.  1), 

/Sf  =  4*7^2 (7) 

V  =  ^'rrH^  =  i'^D- (8.) 

If  we   denote   the   radius  of  a   sphere   by   li,  the   area   of 

any  zone   of  the   sphere   by    S,    its   altitude   by  IT,     and   the 

volume    of   the    corresponding    spherical    sector  by     V,      we 
shaU  have   (P.  X.,  C.  2), 

S  =  2'kE  X  H .     .     .     (9.) 

F  =  |*i22  X  ^ (10.) 

Ii  we  denote  the  volume  of  the  corresponding  spherical 
segment  by  V,  its  altitude  by  H,  the  radius  of  its  ui)per  base 
by  R',  the  radius  of  its  lower  base  by  R",  the  distance  of 
its  upper  base  from  the  centre  by  H',  and  of  its  lower  base 
from  the  centre  by  H",  we  shall  have   (P.  XIV.,   S.) : 

V  ^  ^  •!(  {2  R' X  H  ■{■  R'^  H' ^:  R"  X  I[")     .    .    (11.) 


BOOK    IX. 

SPHERICAL      GEOMETRTj 

DEFINITIONS. 

1.  A  Spherical  Akgle  is  an  angle  included  between  the 
arcs  of  two  great  circles  of  a  sphere  meeting  at  a  point.  The 
arcs  are  called  sides  of  the  angle,  and  their  point  of 
intersection  is  called  the  vertex  of  the  angle. 

The  measure  of  a  sjjherical  angle  is  the  same  as  that  of 
the  diedral  angle  included  between  the  planes  of  its  sides. 
Spherical  angles  may  be  acute,  right,   or  oMuse. 

2.  A  Spherical  PoLYGOisr  is  a  portion  of  the  surface  of 
a  sphere  bounded  by  three  or  more  arcs  of  great  circles. 
The  bounding  arcs  are  called  sides  of  the  polygon,  and  the 
points  in  which  the  sides  meet,  are  called  vertices  of  the 
polygon.  Each  side  is  supposed  to  be  less  than  a  semi-cir- 
cumference. 

Spherical  polygons  are  classified  in  the  same  manner  as 
plane  polygons. 

3.  A  Spherical  Triangle  is  a  spherical  polygon  of  three 
sides. 

Spherical  triangles  are  classified  in  the  same  manner  as 
plane  triangles. 

4.  A  Lune  is  a  portion  of  the  surface  of  a  sphere  bounded 
by  two  semi-circumferences  of  great  circles. 

5.  A  Spherical  Wedge  is  a  portion  of  a  sphere  bounded 
by  a  lune  and  two  semicircles,  which  intersect  in  a  diameter 
of  the  sphere. 


) 


J 


236  GEOMETRY. 

6.  A  Spherical  Pyramid  is  a  portion  of  a  sphere 
bounded  by  a  spherical  polygon  and  sectors  of  circles  whose 
common  centi-e   is  the   centre   of  the   sphere. 

The  spherical  polygon  is  called  the  base  of  the  pyramid, 
and  the  centre  of  the  sphere  is  called  the  vertex  of  the 
pyramid. 

V.  A  PoLB  OP  A  Circle  is  a  point  on  the  surface  of 
the  sphere,  equally  distant  from  all  the  points  of  the  cir 
cumference  of  the   circle. 

8.  A  Diagonal  of  a  spherical  polygon  is  an  arc  of  a 
great  circle  joining  the  vertices  of  any  two  angles  which  are 
not   consecutive. 


PROPOSITION     I.        THEOREM. 

Any  side  of  a  spherical  triangle    is    less    than    the  sum  of 

the  other   two. 

Let  AUC  be  a  spherical  triangle  situated  on  a  sphere 
whose  centre  is  0 :  then  wiU  any  side,  as  AB^  be  less 
than   the   sura  of  the   sides    AG    and    jB(7. 

For,  draw  the  radii  OA^  OB,  and 
0  C :  these  radii  form  the  edges  of  a 
triedral  angle  whose  vertei  is  0,  and 
the  plane  angles  included  between  them 
are  measured  by  the  arcs  AB,  AC, 
and  BC  (B.  m.,  P.  XVIL,  Sch.). 
But  any  plane  angle,  as  A  OB,  is  less 
than  the  sum  of  the  plane  angles  AOC 
and    BOO    (B.  VI.,   P.  XIX.):    hence, 

the    arc    AB    is    less  than    the    sum    of    the   arcs    AC    anl 
BC'y    which  was  to  be  proved. 


BOOK     IX.  237 

C(yr,  1.    Any  side   AB^    of  a  spherical   polygon   ABCDE^ 
is  less  than  the   sum   of  aU  the   other   sides. 

For,  draw  the  diagonals  A  C  and  AD,  dividing  the 
polygon  into  triangles.  The  arc  AB  is  less  than  the  sum 
of  AC  and  BC,  the  arc  AC  is 
less  than  the  sum  of  AD  and  DC, 
and  the  arc  AD  is  less  than  the 
sum  of  DE  and  EA  ;  hence,  AB 
is  less  than  the  sum  of  BC,  CD, 
DE,    and    EA. 

Cor.  2.  The  arc  of  a  small  circle,  on  the  surface  of  a 
sphere,  is  greater  than  the  arc  of  a  great  circle  joining  its 
two  extremities. 

For,  divide  the  arc  of  the '  small  circle  into  equal  parts, 
and  through  the  two  extremities  of  each  part,  suppose  the 
arc  of  a  great  circle  to  be  drawn.  The  sum  of  these  arcs, 
whatever  may  be  their  number,  will  be  greater  than  the  arc 
of  the  great  circle  joining  the  given  points  (0.  1).  But  when 
this  number  is  infinite,  each  arc  of  the  great  circle  will  coin- 
cide with  the  corresponding  arc  of  the  small  circle,  and 
their  sum  is  equal  to  the  entire  arc  of  the  small  circle,  which 
is,   consequently,  greater  than  the  arc  of  the  great  ciicle. 

Cor.  3.  The  shortest  distance  from  one  point  to  another 
on  the  surface  of  a  sphere,  is  measured  on  the  arc  of  a 
great  circle   joining  them. 

PEOPOSITION     n.        THEOREM. 

The  sum   of  the  sides    of   a    spherical  polygon  is  less  than 
the  circumference  of  a  great  circle. 

Let  ABCDE  be  a  spherical  polygon  situated  on  a 
sphere  whose  centre  is  0 :  then  will  the  sum  of  its  sides 
be  less  than  the  circumference  of  a  great  circle. 


238 


GEOMETRY. 


For,  draw  the  radii  OA,  OB,  OC,  OJD,  and  OE: 
these  radii  form  the  edges  of  a  polyedral  angle  whose  vertex 
is  at  0,  and  the  angles  included  between 
them  are  measured  by  the  arcs  AB,  BC, 
CD,  DE,  and  EA.  But  the  s^ira  of 
these  angles  is  less  than  four  right  angles 
(B.  VI.,  P.  XX.)  :  hence,  the  sum  of  the 
arcs  wliich  measure  them  is  less  than  the 
circumference  of  a  great  circle  ;  which  was 
to  he  proved. 


PROPOSITION      HI. 


THEOREM. 


If  a  diameter  of  a  sphere  he  drawn  perpendicular  to  the 
plane  of  any  circle  of  the  sphere,  its  extremities  will  he 
poles  of  that  circle. 

Let  G  be  the  centre  of  a  sphere,  FNG  any  circle  of 
the  sphere,  and  BE  a  diameter  of  the  sphere  perpendicular 
to  the  plane  of  FKG  :  then  will  the  extremities  I)  and  E, 
be  poles   of  the   circle    FNG. 

The  diameter  BE,  being 
perpendicular  to  the  plane  of 
FNG,  must  pass  through 
the  centre  0  (B.  VIII., 
P.  Vn.,  C.  3).  If  arcs  of 
great  circles  BJSF,  BF,  BG, 
•fee,  be  drawn  from  B  to 
different  points  of  the  cir- 
cumference FNG,  and  chords 
of  these  arcs  be  drawn,  these 
chords  will  be  equal   (B.  VI., 

P.  v.),   consequently,  the  arcs   themselves  will  be  equal.      But 
these   arcs  are   the  shortest  lines  that  can  be  dratvn  from  the 


BOOK     IX.  239 

point  2),  to  the  different  points  of  the  circumference  (P.  I., 
C.  2)  :  hence,  the  point  Z>,  is  equally  distant  from  all  the 
points  of  the  circumference,  and  consequently  is  a  pole  of 
the  circle  (D.  7).  In  like  manner,  it  may  be  shown  that 
the  point  E  is  also  a  pole  of  the  circle  :  hence,  both  2), 
and  J5',  are  poles  of  the  circle  FJSTG ;  ichich  was  to  be 
proved. 

Cor.  1.  Let  AMB  be  a  great  circle  perpendicular  to 
BE:  then  will  the  angles  DG3f^  EC 31,  &c.,  be  right 
angles  ;  and  consequently,  the  arcs  DM,  E3I,  &c.,  will 
each  be  equal  to  a  quadrant  (B.  HI.,  P.  XVIL,  S.)  :  hence, 
the  two  poles  of  a  great  circle  are  at  equal  distances  from 
the   circumference. 

Cor.  2.  The  two  poles  of  a  small  circle  are  at  unequal 
distances  from  the  circumference,  the  sum  of  the  distances 
being   equal  to   a  semi-circumference. 

Cor.  3.  If  any  point,  as  M,  in  the  circumference  of  a  great 
circle,  be  joined  with  either  pole,  by  the  arc  of  a  great  circle, 
such  arc  will  be  perpendicular  to  the  circumference  AMB,  since 
its  plane  passes  through  CD,  which  is  perpendicular  to  AMB. 
Conversely:  if  J^/LVbe  perpendicular  to  the  arc  AMB,  it  will  pass 
through  the  poles  D  and  E\  for,  the  plane  of  MN  being  per- 
pendicular to  AMB  and  passing  through  C,  will  contain  CD, 
which  is  perpendicular  to  the  plane  AMB  (B.  VI.,  P.  XVIII.). 

Cor.  4.  If  the  distance  of  a  point  D,  from  each  of  the  points 
A  and  M,  in  the  circumference  of  a  great  circle,  is  equal  to  a 
quadrant,  the  point  D,  is  the  pole  of  the  arc  AM. 

For,  let  C  be  the  centre  of  the  sphere,  and  draw  the 
radii  CD,  CA,  CM.  Since  the  angles  ACD,  MCD,  are 
right  angles,  the  line  CD  is  "jerpendicular  to  the  two 
straight  lines  CA,  CM:  it  is,  therefore,  perpendicular  to  thcii 


240 


GEOMETRY. 


plane   (B.  VI.,  P.  IV.)  ;    hence,  tlie  point    Z>,    is  the  pole  of 
the   arc    A3I. 

Scholium.  The  properties  of  these  poles  enable  ns  to 
describe  arcs  of  a  cii'cle  on  the  surface  of  a  sphere,  with 
the  same  facility  as  on  a  plane  surface.  For,  by  turning 
the  arc  DF  about  the  point  2>,  the  extremity  F  Trill 
describe  the  small  circle  FNG ;  and  by  turning  the  quad- 
rant DFA  round  the  point  J),  its  extremity  A  will 
describe   an   arc   of  a  great   circle. 


PE0P0SITI0T7    rv. 


THEOREM. 


The  angle  formed  hy  two  arcs  of  great  circles^  is  equal  to 
that  formed  by  the  tangents  to  these  arcs  at  their  point 
of  intersection^  and  is  m,easured  by  the  arc  of  a  great 
circle  described  from  the  vertex  as  a  pole^  and  limited 
by  the  sides,  produced  if  necessary. 

Let  the  angle  BAG  be  formed  by  the  two  arcs  AB, 
AC:  then  is  it  equal  to  the  angle  FAG  formed  by  the 
tangents  AF,  AG,  and  is  measured  by  the  arc  DF  of 
a  great  circle,  described  about    ^    as  a  pole. 

For,  the  tangent  AF,  drawn  in  the 
plane  of  the  arc  AB,  is  perpendicular 
to  the  radius  A  0 ;  and  the  tangent 
AG,  drawn  in  the  plane  of  the  arc 
AC,  is  perpendicular  to  the  same  radius 
A  0 :  hence,  the  angle  FA  G  is  equal 
to  the  angle  contained  by  the  j)lanes 
ABBH,  ACEH  (B.  VI.,  D.  4)  ;  which 
is  that  of  the  arcs  AB,  A  C.  Now,  if 
the  arcs  AD  and  AF  a/e  both  quad- 
rants, the    lines    OB,     OF,    are    perpendicular  to     OA,    and 


BOOK     IX. 


241 


the  angle  DOE  is  equal  to  the  angle  of  the  planes  ABDH^ 
ACJEII:  hence,  the  arc  DE  is  the  measure  of  the  ancle 
contained  by  these  planes,  or  of  the  angle  CAB ;  which 
teas  to  be  proved. 

Cor.  1.    The    angles    of   spherical    triangles    may   be   com 
pared  by   means   of   the   arcs   of   great   circles   described   from 
their  vertices   as   poles,   and   included  between   their   sides. 

A  spherical    angle    can    always    be    constructed   equal  to   a 
given   Bpherical   angle. 

Cor.  2.  Vertical  angles,  such  as 
ACQ  and  BC.W  are  equal;  for 
either  of  them  is  the  angle  formed 
by  the  two  phnes  ACB,  OCN. 
When  two  arcs  ACB,  OCJST,  in- 
terse ct,  the  sum  oi  t^o  adjacent 
angles,  as  AGO,  OCJS,  is  equal 
to   two   right   angles. 


PROPv:)sinoiT    V. 


THEOREM. 


J^  from  the  vertices  of  the  angles  of  a  spherical  triangle., 
as  poles,  arcs  be  dcstirlbed  forming  a  spherical  triangle^ 
the  vertices  of  the  angles  of  this  second  triangle  will  be 
respectively  poles  of  the  sides   of  the  first. 

From  the  vertices  4,  B,  (7, 
as  poles,  let  the  arcs  EF,  ED, 
ED,  be  described,  forming  the 
tiiangrle  BEE:  then  will  the 
vertices  B,  E,  and  F,  be 
respectively  poles  of  the  sides 
BC,    AC,    AB. 

For,    the     point     A      being 

16 


242 


GEOMETRY. 


the  pole  of  the  arc  EF^  the  distance  AE,  is  a  quadrant ; 
the  point  C  being  the  pole  of  the  arc  JDE^  the  distance 
CE^  is  likewise  a  quadrant :  hence,  the  point  ^  is  at  a 
quadrant's  distance  from  the  points  A  and  C :  hence,  it  ia 
the  pole  of  the  arc  A  C  (P.  III.,  C.  4).  It  may  be  shown, 
in  like  manner,  that  D  is  the  pole  of  the  arc  .BC\  and 
E    that   of  the   arc    AB ;    which   was   to   be  proved. 

Scholium.     The    triangle  ABC,      may    be     described    by 

means   of    DEE,    as    DEE  is  described  by  mctans   of   ABC. 

Triangles  thus  related  are  called  polar  triangles,  or  supple- 
mental triangles. 

PROPOSITION      VI.        THEOEEM. 


Any  angle,  in  one  of  two  polar  triangles,  is  measured  hjj  a 
semi-circumference,  minus  the  side  lying  op>2yosite  to  it  in 
the   other   triangle. 

Let  ABC,  and  EED,  be  any  two  polar  triangles: 
then  will  any  angle  in  either  triangle  be  measured  by  a 
semi-circumference,  minus  the  side  lying  opposite  to  it  in  the 
other   triangle. 

For,  produce  the  sides  AB, 
A  C,  if  necessary,  till  they 
meet  EF,  in  6^  and  II.  The 
point  A  being  the  pole  of 
the  arc  GH,  the  angle  A  is 
measured  by  that  arc  (P.  IV.). 
But,  since  E  is  the  pole  of 
All,  the  arc  EII  is  a  quad- 
rant ;     and    since     F    is     the 

pole    of   AG,    EG    is   a   quadrant  :    hence,   the    sum    of  the 
arcs    EH    and     GF,    is    equal   to   a   semi-circumference.      But. 


BOOK     IX 


243 


the  sura  of  the  arcs  EII  and  GF^  is  equal  to  the  sum 
of  the  arcs  EF  and  Gil :  hence,  the  arc  GII^  which 
measures  the  angle  .1,  is  equal  to  a  semi-circumference, 
minus  the  arc  EF.  In  Uke  manner,  it  may  be  shown,  that 
any  other  angle,  in  either  triangle,  is  measured  by  a  semi- 
circumference,  minus  the  side  lying  opposite  to  it  in  thj 
ether  triangle  ;    which  teas   to   be  proved. 


Scholhcm.  Besides  the  triangle  DEF, 
three  others  may  be  formed  by  the  inter- 
section of  the  arcs  DE,  EF,  EF.  But 
the  proposition  is  applicable  only  to  the 
central  triangle,  which  is  distinguished 
from  the  other  three  by  the  circumstance, 
that  the  two  vertices,  A  and  D,  lie  on  the 
same  side  oi  BC;  the  two  vertices,  B  "  " 
and  E,  on  the  same  side  of  AC;  and 
the   two  vertices,    C  and   F,   on   the  same   side   of  AB. 


PROPOSITION     YII. 


TiiEORE:\r. 


rf  from  (he  vertices  of  any  two  angles  of  a  spherical  tri- 
angle^ as  poles ^  arcs  of  circles  be  described  passing 
through  the  vertex  of  the  third  angle  /  and  if  from  the 
second  point  in  which  these  arcs  intersect^  arcs  of  great 
circles  be  drawn  to  the  vertices,  used  as  poles,  the  parts 
of  the  triangle  thus  formed  will  be  equal  to  those  of  the 
given   triangle,   each   to  each. 

Let  ABC  be  a  spherical  triangle  situated  on  a  spliere 
whose  centre  is  0,  CED  and  GFD  arcs  of  circles 
described  about  B  and.  A  as  poles,  and  let  DA  and 
DB    be   arcs   of    great   circles  :    then    will    the    parts    of   the 


2t4 


GEOMETRY. 


triangle     ABD     be    equal    to    those    of   tlie    giveu    triangle 
ABC^    each  to   each. 

For,  by  construction,  the  side  AD 
is  equal  to  A  C,  the  side  BB  is 
?qual  to  BC^  and  the  side  AB  is 
rommon  :  hence,  the  sides  are  equal, 
each  to  each.  Draw  the  radii  OA^ 
OB,  OC,  and  OB.  The  radii  OA, 
OB,  and  OC,  will  form  the  edges 
of  a  triedral  angle  whose  vertex  is 
0 ;    and    the    radii     OA,     OB,     and 


will    form  the 


edges  of  a  second  triedral  angle  whose  vertex  is  also  at  0  ; 
and  the  plane  angles  formed  by  these  edges  will  be  equal, 
each  to  each :  hence,  the  planes  of  the  equal  angles  are 
equally  inchned  to  each  other  (B.  VI.,  P.  XXI.).  But,  the 
angles  made  by  these  planes  are  equal  to  the  corresponding 
spherical  angles ;  consequently,  the  angle  BAD  is  equal  to 
BAG,  the  angle  ABD  to  ABC,  and  the  angle  ADB 
to  A  CB :  hence,  the  parts  of  the  triangle  ABD  are  equal 
to  the  parts  of  the  triangle  A  CB,  each  to  each ;  which 
was  to  be  proved. 


Scholium  1.  The  triangles  ABC  and  ABD,  are  not, 
in  general,  capable  of  superposition,  but  their  parts  are 
symmetrically  disposed  with  respect  to  AB.  Triangles  which 
have  all  the  parts  of  the  one  equal  to  all  the  parts  of  the 
■ether,  each  to  each,  hut  not  capable  of  superposition,  are 
called,   symmetrical  triangles. 

Scholium  2,  If  symmetrical  triangles  are  isosceles,  they 
can  be  so  placed  as  to  coincide  throughout :  hence,  they  are 
Bq^lal  in  area. 


BOOK     IX.  245 


PROPOSITION      VIII.        THEOREM. 

If  two  spJierical  triangles^  on  the  same,  or  on  equal  S2)heresy 
have  two  sides  and  the  included  ayigle  of  the  one  equal 
to  two  sides  and  the  included  angle  of  the  other,  each 
to  each,   the  remainijig  parts  are  equal,  each  to   each. 

Let  the  spherical  triangles  ABC  and  EFG,  have  the 
side  EF  equal  to  AB,  the  side  EG  equal  to  AG,  and 
the  angle  FEG  equal  to  BAG:  then  will  the  side  FG  be 
equal  to  BC,  the  angle  EFG  to  ABC,  and  the  angle 
EGF   to    ACB. 

For,  the  triangle  EFG  may- 
be placed  upon  ABC,  or  upon 
its  symmetrical  triangle  ABB,  so 
as  to  coincide  with  it  throughout, 
as  may  be  shown  by  the  same  j)^  j  \q  q^ 
course  of  reasoning  as  that  em- 
ployed in  Book  I.,  Proposition  V.  : 
hence,   the    side    FG    is    equal  to 

BG^    the  angle    EFG    to    ABC,    and  the  angle    EGF    to 
A  CB ;    which  was  to  be  proved. 


PROPOSITION      IX.        THEOREM. 

If  two  spherical  triangles  on  the  same,  or  on  equal  spheres, 
have  two  angles  and  the  included  side  of  the  one  equal 
to  two  angles  and  the  included  side  of  the  other,  each 
to  each^   the  remaining  parts  will  be  equal,  each  to  each 

Let  the  spherical  triangles  ABC  and  EFG,  have  the 
angle  FEG  equal  to  BAG,  the  angle  EFG  equal  to 
ABC,    and  the  side     EF    equal    to     AB :     then    will    the 


246 


GEOMETRY. 


side     EG    be    equal    to    AC,    the    side    FG    to    BC,    and 
the   angle    FGE    to    BOA. 

For,  the  triangle  EFG  may 
be  placed  upon  ABC,  or  upon 
its  symmetrical  triangle  ABB,  so 
as  to  coincide  with  it  throughout, 
as  may  be  shown  by  the  same 
course  of  reasoning  as  that  em- 
ployed in  Book  I.,  Proposition 
VI. :  hence,  the  side  FG  is  equal 
to  AC,  the  side  FG  to  BC, 
BCA  ;    which  teas   to   be  proved. 


C  G 


and  the  angle    FGE    to 


PROPOSITION      X.        THEOREM. 

If  two  spherical  triangles  on  the  same,  or  on  equal  spheres, 
have  their  sides  equal,  each  to  each,  their  angles  will  he 
equal,  each  to  each,  the  equal  angles  lying  oj^posite  the 
equal    sides. 

Let  the  spherical  triangles  EFG  and  ABC  have  the 
side  EF  equal  to  AB,  the  side  EG  equal  to  AC,  and 
the  side  FG  eqixal  to  BC:  then  will  the  angle  FEG  be 
equal  to  BAC,  the  angle  EFG  to  ABC,  and  the  angle 
EGF  to  ACB,  and  the  equal  angles  will  lie  opposite  the 
equal   sides. 

For,  it  may  be  shown  by  the 
same  course  of  reasoning  as  that 
employed  in  B.  I.,  P.  X.,  that  the 
triangle  EFG  is  equal  in  aU 
respects,  either  to  the  triangle 
ABC,  or  to  its  symmetrical  tri- 
angle ABB  :  hence,  the  angle 
FEG,   opposite  to  the  side    FG,    is  equal  to  the  angle    BAC, 


10  G 


BOOK     IX. 


247 


opi)osite  to  BC i  the  angle  EFG^  opposite  to  EG^  is  equal 
to  the  angle  ABC,  opposite  to  AC \  and  the  angle  EGF, 
oj-posite  to  EF,  is  equal  to  the  angle  A  CJ5,  opposite  to 
A  B  ;    which  teas   to   be  proved. 


PROPOSITION      XI.        THEOREM. 

Li  any  isosceles  spherical  ti'iangle,  the  angles  opposite  the 
equal  sides  are  equal ;  and  conversely,  if  tico  angles  of 
a  spherical  triangle  are  equal,   the  triangle  is  isosceles. 

1°.  Let  ABC  be  a  spherical  triangle,  having  the  side 
AB  equal  to  AC :  then  will  the  angle  C  be  equal  to 
the   angle    B. 

For,  draw  the  arc  of  a  great  circle 
from  the  vertex  A,  to  the  middle  point 
Z>,  of  the  base  B  C :  then  in  the  two 
triangles  ABB  and  ABC,  we  shall  have 
the  side  AB  equal  to  AC,  by  hypothe- 
sis, the  side  BB  equal  to  BC,  by  con- 
struction, and  the  side  AB  common  ; 
consequently,  the  triangles  have  their  angles  equal,  each  to 
each  (P.  X.)  :  hence,  the  angle  C  is  equal  to  the  angle 
B  ;    which  was   to   be  p^roved. 

2°.  Let  ABC  be  a  spherical  triangle  having  the  angle 
6^  equal  to  the  angle  B  :  then  \vill  the  side  AB  be 
e(iual  to  the  side  AC,  and  consequently  the  triangle  wil! 
be   isosceles. 

For,  suppose  that  AB  and  A  C  are  not  equal,  but  that 
one  of  them,  as  AB,  is  the  greater.  On  AB  lay  off  the 
arc  BO  equal  to  AC,  and  draw  the  arc  of  a  great  circle 
from  0  to  C  :  then  in  the  triangles  ACB  and  OBC, 
we   sliall  have   the  side    AC    eqial   to    OB,    by  construction, 


248 


GEOMETRY. 


the  side    BC    common,  and  the   included    angle    ACB    equal 
to    the    included    angle     OBC^     by    hypothesis  ,     hence,    the 
remaining   parts   of  the   triangles   are    equal, 
each   to   each,   and    consequently,   the   angle 
OC-B    is   equal  to  the   angle    ABC.      But, 
the   angle    ACB     is   equal    to    ABC,     by 
hypothesis,    and   therefore,    the    angle     OCB 
is   equal   to   A  CB,      or   a    part   is   equal   to 
the   whole,   which   is  imjiossible  :    hence,   the 
supposition    that    ^17?      and     A  C     are    un- 
equal,  is   absurd  ;    they  are   therefore   equal,  and   consequently, 
the   triangle    ABC    is  isosceles  ;    which  was   to  he  proved. 

Cor.  The  triangles  ABB  and  ABC,  having  all  of 
their  parts  equal,  each  to  each,  the  angle  ABB  is  equal 
to  AB  C,  and  the  angle  BAB  is  equal  to  BA  C  ;  that 
is,  if  an  arc  of  a  great  circle  be  drawn  froin  the  vertex. 
of  an  isosceles  spherical  triangle  to  the  middle  of  its  base, 
it  will  be  perpendicular  to  the  base,  and  will  bisect  the  verti- 
cal angle  of  the   triangle. 

PROPOSITION'     XII.        THEOEEM. 


In  any  spherical  triangle,  the  greater  side  is  opj)osite  the 
greater  angle  ;  and  conversely,  the  greater  angle  is  oppo- 
site the  greater  side. 

1°.  Let  ABC  be  a  spherical  triangle,  in  which  the  angle 
A  IS  greater  than  the  angle  B  :  then  will  the  side  BO 
be  greater  than  the  side  AC. 
For,  draw  the  arc  AB, 
making  the  angle  BAB  equal 
to  ABB :  then  will  AB  be 
equal  to  BB  (P.  XI.).  But, 
the   sum  of   AB    and    BG    is 


B  O  O  K     I X  .  249 

greater  than  AC  (P.  I.)  ;  or,  putting  for  AD  ita  equal 
J^J),  we  have  the  sum  of  J?Z>  aud  DC,  or  DC,  greater 
than    AC',    which  teas   to  be  proved. 

2°.  In  the  triangle  ADC,  let  the  side  i>(7  be  greater 
than  AC:  then  will  the  angle  A  be  greater  than  the 
angle    D. 

For,  if  the  angles  A  and  D  were  equal,  the  sides  DC 
and  A  C  would  be  equal  ;  or  if  the  angle  A  was  less 
than  the  angle  D,  the  side  DC  would  be  less  than  AC, 
either  of  which  conclusions  is  contrary  to  the  hypothesis:  hence, 
the  angle  A  is  greater  than  the  angle  D ;  which  ivas  to  he  proved. 


PEOPOSITILX     XIII.        THEOREM. 

If    two    triangles    on    the    same,    or    on    equal    spheres,    are 
mutually  equiangular,   they   are    also    mutually  equilateral. 

Let  the   spherical  triangles    A     and    D,    be  mutually  equi- 
angular :    then  will  they   also   be   mutually  equilateral. 

For,  let  D  be  the  polar  triangle  of  A, 
and  Q  the  polar  triangle  of  D  :  then,  be- 
cause the  triangles  A  and  D  are  mutually 
equiangular,  their  polar  triangles  P  and  Q, 
must  be  mutually  equilateral  (P.  VI.),  and  con- 
sequently mutually  equiangular  (P.  X.).  But, 
the  triangles  P  and  Q  being  mutually  equi- 
angular, their  polar  triangles  A  and  D,  are 
mutually   equilateral   (P.  VI.)  ;    which  was   to   he  proved. 

Scholium.    This  proposition   does  not   hold   good   for  plane 
triangles,    for    all    similar    plane    triangles    are    mutually  equi- 
angular,    but     not     necessarily     mutually     equilateral.       Two 
spherical  triangles  on   the   same  or  on   equal   spheres,   cannot 
be   similar  witliout  being  equal  in   all  their  parts. 


250 


GEOMETRY. 


PKOPOSITION     XIV.         THEOEEM. 

The  sum   of  the   angles   of  a   spherical  triangle  is   less    than 
six  right   angles,   and  greater  than   two  right   angles. 

Let    ABC    be   a    spherical    triangle,   and    DEF    its   polar 
triangle  :    then    o'ill   the   sum  of   the   angles    A,     J3,     and     (7,, 
be   less   than   six   right   angles  and  greater   than    two. 

For,  any  angle,  as  A,  be- 
ing measured  by  a  senii-cii*- 
cumference,  minus  the  side 
EF  (P.  VI.),  is  less  than  two 
right  angles :  hence,  the  sum 
of  the  three  angles  is  less  than 
six  right  angles.  Again,  be- 
cause the  measure  of  each  angle 
is  equal  to  a  semi-circumference 
minus  the   side   lying   opposite 

to  it,  in  the  polar  triangle,  the  measure  of  the  sum  of  the 
three  angles  is  equal  to  three  semi-circumferences,  minus  the 
sum  of  the  sides  of  the  polar  triangle  DEF.  But  the 
latter  sum  is  less  than  a  circumference  ;  consequently,  the 
measure  of  the  sum  of  the  angles  A,  B,  and  C,  is 
greater  than  a  semi-circumference,  and  therefore  the  sum  of 
the  angles  is  greater  than  two  right  angles  :  hence,  the  sura 
of  the  angles  A,  7?,  and  C,  is  less  than  six  right  angle*, 
and   greater  than   two  ;    which  was   to  'he  proved. 

Cor.  1.  The  sum  of  the  three  angles  of  a  spherical  tri- 
angle is  not  constant,  like  that  of  the  angles  of  a  rectilineal 
tri-angle,  but  varies  between  two  right  angles  and  six,  \\nth- 
out  ever  reaching  either  of  these  limits.  Two  angles,  there- 
fore,  do  not   serve   to   determine   the   third. 


BOOK     IX.  251 

Cor.  2.  A  spherical  triangle  may  have  two,  or  even  three 
of  its  angles  riglit  angles ;  also  two,  or  even  three  of  its 
au'T'les   obtuse. 

O 

Cor.  3.  If  a  triangle,  ABC,  is  hi-rectangular, 
that  is,  has  two  riglit  angles  B  and  C,  the  vertex 
A  will  be  the  pole  of  the  other  side  BC,  and 
AB,  AC,  will   be   quadrants. 

For,  since  the  arcs  AB  and  AC  are  perpen- 
dicular   to    BC,    each    must    pass    through    its 
pole  (P.  III.,  Cor.  3) :   hence,  their  intersection  A  is  that  pole, 
and   consequently,   AB  and   AG  are   quadrants. 

If  the  angle  A  is  also  a  right  angle,  the  triangle  ABC 
is  tri-rectangular ;  each  of  its  angles  is  a  right  angle,  and 
its  sides  are  quadrants.  Four  tri-rectangular  triangles  make 
up  the  surface  of  a  hemisphere,  and  eight  the  entire  surface 
of  a   sphere. 

ScJioUum.  The  right  angle  is  taken  as  the  unit  of  mear 
sure   of  spherical   angles,   and   is   denoted   by    1. 

The  excess  of  the  sum  of  the  angles  of  a  spnerical  tri- 
angle over  two  right  angles,  is  called  the  spherical  excess. 
II  we  denote  the  spherical  excess  by  £J,  and  the  three 
angles  expressed  in  terms  of  the  right  angle,  as  a  unit,  by 
Aj    -Z?,     and    C,     we   shall  have,  • 

U  =  A  +  B  +  C  -  2. 

The  spherical  excess  of  any  spherical  polygon  is  equal  to 
the  excess  of  the  sum  of  its  angles  over  two  right  angles 
taken  as  many  times  as  the  polygon  has  sides,  less  two. 
If  we  denote  the  spherical  excess  by  ^,  the  sum  of  the 
angles  by  S,  and  the  number  of  sides  by  n,  we  shall 
have, 

E  =  S  -  2{n  -  2)   =  aS  -  2/1  +  4. 


i52 


GEOMETRY. 


PROPOSITION     XV.        THEOREM. 


Any  lune,  is  to  the  surface  of  the  sphere,  as  the  arc  lohicU 
measures  its  angle  is  to  the  circumference  of  a  great 
circle ;  or,  as  the  angle  of  the  lune  is  to  four  rigid 
angles. 

Let  AMBN  be  a  lune,  and  MCN  the  angle  of  the  lune, 
then  will  the  area  of  the  lune  be  to  the  surface  of  the  sphere, 
as  the  arc  MN  is  to  the  circumference  of  a  great  circle 
MNPQ ;  or,  as  the  angle  MCN  is  to  four  right  angles 
(B.  Ill,   P.  XVII.,   C.  2). 

In  the  first  place,  supjDose  the  arc 
^IN  and  the  circumference  MNFQ 
to  be  commensurable.  For  example, 
let  them  be  to  each  other  as  5  is 
to  48.  Divide  the  circumference 
MNPQ  into  48  equal  parts,  be- 
ginning at  M ;  MN  will  contain 
five   of  these   parts.      Join  each  point 

of  division  with  the  points  A  and  '-B,  by  a  quadrant : 
there  wiU  be  formed  96  equal  isosceles  spherical  triangles 
(P.  Vil.,  S.  2)  on  the  surface  of  the  sphere,  of  Mhich  the 
lune  will  contain  10  :  hence,  in  this  case,  the  area  of  the 
lune  is  to  the  sui-face  of  the  sphere,  as  10  is  to  96,  or 
as  5  is  to  48  ;  that  is,  as  the  arc  MN  is  to  the  circura- 
ference  3INPQ,  or  as  the  angle  of  the  lune  is  to  fif^r 
riglit   angles. 

In  like  manner,  the  same  relation  may  be  shown  tQ 
exist  when  the  arc  MN,  and  the  circumference  MNPQ 
are  to   each   other   as   any   other   whole  numbers. 

If  the   arc    MN,    and  the  circumference    MNPQ,    are  not 
C5ommensurable,  the   same   relation   may  be   shown   to   exist   by 


I 


BOOK    IX.  253 

a  course  of  reasoning  entirely  analogous  to  that  employed 
in  Book  IV.,  Proposition  III.  Ilence,  in  all  cases,  the  area 
of  a  lune  is  to  the  surface  of  the  sphere,  as  the  arc  meas- 
uring the  angle  is  to  the  circumference  of  a  great  circle; 
or,  as  the  angle  of  the  lune  is  to  four  right  angles  ;  which 
was   to  be  2^fOved. 

Cor.  1.  Luncs,  on  the  same  or  on  equal  sj^heres,  are  to 
each   other  as   theu-   antjles. 

Cor.  2.  If  we  denote  the  area  of  a  tri-rectanfrular  triansrle 
by  T,  the  area  of  a  lune  by  i,  and  the  angle  of  the 
lune  by  A^  the  right  angle  being  denoted  by  1,  we  shall 
have, 

L    :     ST    :  :    A    :    4  I 
whence, 

L  =  Tx  2A  i 

hence,  the  area  of  a  lune  is  eqiial  to  the  area  of  a  tn- 
rectangular  triangle  multiplied  by  twice  the  angle  of  the 
lune.    , 

Scholium.  The  spherical  wedge,  whose  angle  is  MGIT^ 
is  to  the  enth-e  sphere,  as  the  angle  of  the  wedge  is  to  four 
right  angles,  as  may  be  shown  by  a  course  of  reasoning 
entirely  analogous  to  that  just  emj^loyed :  hence,  we  infei 
that  the  volume  of  a  spherical  wedge  is  equal  to  the  lune 
which   forms  its  base,   multipHed  by    one-thu'd   of  the   radius. 


PROPOSITION      XYI.        THEOREM. 
Symmetrical  triangles  are  equal  in  area. 

Let  ABC  and  BEF  be  symmetrical  triangles,  the 
side  DE  bemg  equal  to  AB,  the  side  DF  to  AC^  and 
the   side    EF    to    BC  '.    then   will   the   triangles    be   equal   in 


254 


GEOMETRY. 


For,  conceive  a  small  circle  to  be  dia'\m  through  A^  B^ 
and  C,  and  let  P  be  its  pole  ;  draw  arcs  of  great  circles 
from  P  to  A^  P,  and  C:  these 
arcs  will  be  equal  (D.  V).  Draw 
the  arc  of  a  great  circle  P'Q, 
making  the  angle  PFQ  equal  to 
AGP,  and  lay  off  on  it,  FQ 
equal  to  GP\  draw  arcs  of  great 
circles   QD    and     QE. 

In     the     triangles     PAG       and 
FDQ,      we     have     the     side     FD 

equal  to  A  C,  by  hypothesis ;  the  side  FQ  equal  to  PG 
by  construction,  and  the  angle  DFQ  equal  to  AGP,  by 
construction  :  hence  (P.  VIIL),  the  side  DQ  is  equal  to 
AP,  the  angle  FDQ  to  PAG,  and  the  angle  FQD  to 
APG.  Now,  because  the  triangles  QFD  and  PAG  are 
isosceles  and  equal  in  all  their  parts,  they  may  be  placed  so 
as  to  coincide  throughout,  the  base  FD  falling  on  AG^ 
DQ  on  GP,  and  FQ  on  AP:   hence,  they  are  equal  in  area. 

If  we  take  from  the  angle  DFE  the  angle  DFQ,  and 
from  the  angle  xiCB  the  angle  AGP,  the  remaining 
angles  QFE  and  PCB,  will  be  equal.  In  the  triangles 
FQE  and  PCB,  we  have  the  side  QF  equal  to  PG, 
by  construction,  the  side  FE  equal  to  BC,  by  hypothesis, 
and  the  angle  QFE  equal  to  PGB,  from  what  has  just 
been  shown  :  hence,  the  triangles  are  equal  in  all  theii 
parts,  and  being  isosceles,  they  may  be  placed  sj  as  t^ 
coincide  throughout,  the  side  QE  falling  on  PG,  a,nd  th^ 
side  QF  on  PB ;  these  triangles  are,  therefore,  equal  m 
area. 

In  the  triangles  QDE  and  PAB,  we  have  the  sides 
QD,  QE,  PA,  and  PB,  all  equal,  and  the  angle  DQE 
equal  to  APB,  because  they  are  the  sums  of  equal  angles: 
hence,     the     triangles     are     equal     in     all     their    parts,    and 


BOOK     IX.  255 

because  they  are  isosceles,  they  may  be  so  placed  as  to 
coincide  throughout,  the  side  QD  falling  on  PB^  and  the 
side  QE  on  PA  ;  these  triangles  are,  therefore,  equal  in 
srea. 

Hence,  the  sura  of  the  triangles  QFD  and  QFE^  is 
ejual  to  the  sum  of  the  triangles  PAC  and  PBC.  If 
from  the  former  sum  we  take  away  the  triangle  QDE^ 
there  will  remain  the  triangle  DFE\  and  if-  from  the  latter 
sura  we  take  away  the  triangle  PAB^  there  Avill  remain 
the  triangle  ABC  ;  hence,  the  triangles  ABC  and  DEF 
are   equal   in   area  ;    which   icas   to   be  proved. 

Scholium.  If  the  point  P  falls  within  the  triangle  ABC, 
the  point  Q  will  fall  within  the  triangle  BEE.  In  this 
case,  the  triangle  BEE  is  equal  to  the  sum  of  the  triangles 
QEB,  QEE,  and  QBE,  and  the  triangle  ABC  is  equal 
to  the  sum  of  the  equal  triangles  PAC,  PBC,  and  PAB' 
the   proposition,   therefore,   still   holds  good. 

PROPOSITIO]^     XVII.        THEOREM. 

If  the  circumferences  of  tiDO  great  circles  intersect  on  the 
surface  of  a  hemisphere,  the  sum  of  the  opposite  triangles 
thus  formed,  is  equal  to  a  lu?ie  whose  angle  is  equal  to 
that  formed  by   the  circles. 

Let  the  circumferences  A  OB,  COD, 
intersect  on  the  surface  of  a  hemis- 
phere :  then  will  the  sum  of  the  oppo- 
site triangles  AOC,  BOB,  be  equal 
to  the   lune   whose   angle   is    BOB. 

For,  produce  the  arcs  OB,  OB, 
on  the  other  hemisphere,  till  they  meet 
at  iV:  Now,  since  A  OB  and  OBJST 
are    semi-circumferences,    if  we    take    away   the   common   part 


256 


GEOMETRY. 


OB^  we  shall  have  BN  equal  to  AO.  For  a  like  rea- 
son, we  have  DN  equal  to  CO^  and  BD  equal  to  AC x 
hence,  the  two  triangles  AOC,  BDIT^ 
have  their  sides  respectively  equal : 
they  are  therefore  symmetrical  ;  con- 
sequently, they  are  equal  in  area 
(P.  X\a.).  But  the  sum  of  the  tri- 
angles BBJST,-  BOB,  is  equal  to 
the  lune  OBNDO,  whose  angle  is 
B OB  :  hence,  the  sum  of  AOG  and 
BOB  is  equal  to  the  lune  Avhose 
angle  is    B  OB ;    wliicli  was   to   he  proved. 

Scholhim.  It  is  evident  that  the  two  spherical  pyramids, 
which  have  the  triangles  AOC,  BOB,  for  bases,  are 
together  equal   to  the   spherical  wedge  whose   angle  is    BOB. 


PKOPOSITION      XVIII. 


THEOEEM. 


The  area  of  a  spherical  triangle    is    equal    to    its    spherical 
excess  multiplied  by  a  tri-rectangidar  triangle. 

Let    ABC    be   a   spherical  triangle  ;    then   will   its   surface 

be   equal   to 

(^  +  ^  +  (7  -  2)  X   T. 

For,  produce  its  sides  till  they  meet 
the  great  circle  BEFG,  drawn  at  plea- 
sure, without  the  triangle.  By  the  last 
theorem,  the  two  triangles  ABE,  AGH, 
are  together  equal  to  the  lune  whose 
Rngle  is  A  ;  but  the  area  of  this  lune 
is    equal    to     2.4  x  T     (P.  XY.,   C.  2)  : 

hence,   the   sum   of  the   triangles    ABE    and    AGII,    is   equal 
to     lA  X  T.      In   like    manner,   it  may    be    shown    that    the 


B  O  0  K     I X .  257 

siun  of  the  triangles  BFG  and  BID^  is  equal  to  2B  x  T, 
and  that  the  sura  of  the  triangles  CIII  and  CI^£J,  is 
equal   to     20  X  T. 

But  the  sum  of  these  six  triangles  exceeds  the  hemis- 
phere, or  foiir  times  T,  by  twice  the  triangle  ABC.  We 
shall  therefore   have, 

2  X  area  ABC  =  2A  x  T -\-  2B  x  T-\-  2C  x  T  -  \T  \ 
or,   by   reducing   and   factoring, 

area  ABC  =  {A  +  B  -\-  C  -  2)  x  T; 
which  loas  to  he  proved. 

Scholhtm  1.  The  same  relation  which  exists  between  the 
spherical  triangle  ABC^  and  the  tri-rectangular  triangle, 
exists  also  between  the  spherical  pyi-amid  which  has  ABC 
for  its  base,  and  the  tri-rectangular  pyi-amid.  The  triedral 
angle  of  the  pp-amid  is  to  the  triedral  angle  of  the  tri- 
rectangular  pyramid,  as  the  triangle  ABC  to  the  tri-rectan- 
gular triangle.  From  these  relations,  the  following  conse- 
quences  are   deduced  : 

1°.  Triangular  spherical  pyramids  are  to  each  other  as 
their  bases ;  and  since  a  polygonal  pyramid  may  always  be 
di\'ided  trto  triangular  pyramids,  it  follows  that  any  two 
spherical    pyramids   are   to   each   other   as   their   bases. 

2°.  Polyedral  angles  at  the  centre  of  the  same,  or  of 
equal  spheres,  are  to  each  other  as  the  spherical  polygons 
Intercepted   by   their   faces. 

Scholium  3.  A  triedral  angle  whose  feces  are  perpen- 
dicular to  each  other,  is  called  a  rigid  triedral  angle ; 
and  if  the  vertex  be  at  the  centre  of  a  sphere,  its  faces  will 
intercept  a  tri-rectangular  triangle.  The  right  triedi-al  angle  is 


258  GEOMETRY. 

taken   as  tlie  uiiit   cf  polyedial   angles,  aud  the  tri-rectangular 

sijlierical   triangle   is   taken   as   its   measure.  If  the    vertex    of 

a   polyedral   angle  be  taken   as   the    centre  of    a    sjihere,    the 

portion   of    the   surface   intercepted    by   its  faces   will     be    the 
measure   of   the    polyedral   angle,   a   tri-reetangular   triangle   ol 
Lo   same    sphere,   being   the   unit. 


PROPOSITION      XIX.        TIIEOPEM. 

The   area   of  a    spherical    'polygon    is    equal  to   its   s2->herical 
excess  midtijjlied  by  tJie  tri-reetangular  triangle. 

Let  ABCDE  be  a  spherical  polygon,  the  sum  of  whose 
angles  is  /S,  and  tlie  number  of  whose  sides  is  n  :  tlieu 
win   its   area   be   equal    to 

(^  _  2w  +  4^  X   T. 

For,  draw  the  diagonals  AC,  AD, 
dividing  the  polygon  into  spherical  tri- 
angles :  there  will  he  n  —  2  such  tri- 
ano'les.  Now,  the  area  of  each  tri- 
angle  is  equal  to  its  spherical  excess 
into  the  tri-rectangular  triangle  :   hence, 

the  sura  of  the  areas  of  all  the  triangles,  or  the  area  of  the 
polygon,  is  equal  to  the  sum  of  all  the  angles  of  the  tri- 
angles, or  the  sum  of  the  angles  of  the  polygon  diminished 
by    2{n  —  2)     into  the   tri-rectangular  triangle  ;     or, 


area  ABCDE  =  [S  -  2(n  -  2)]  x  T 
whence,   by   reduction, 

area  ABCDE  =  {^S  ~  2n  ■\-  A)  x  T  \ 

which  was  to  be  proved. 


BOOK    IX.  259 


GENERAL   SCHOLIUM. 

From  any  point  on  a  hemispliere,  two  arcs  of  a  great  cir- 
rle  can  always  be  drawn  which  shall  be  perj^ondicular  to  the 
circumference  of  the  base  of  the  hemisphere,  and  they  will  in 
general  be  unequal.  Now,  it  may  be  proved,  by  a  course  of 
reasoning  analogous  to  that  employed  in  Book  L,  Proposition 
XV.: 

1°.  That  the  shorter  of  the  two  arcs  is  the  sliortost  arc 
that  can  be  drawn  from  the  given  point  to  the  circum- 
ference . 

2°.  That  two  obhqne  arcs  drawn  from  the  same  point,  to 
points  of  the  circumference  at  equal  distances  from  tlie  foot 
of  the    perpendicular,    are   equal  : 

3°.  That  of  two  oblique  arcs,  that  is  the  longer  which 
meets  the  circumference  at  the  greater  distaar*  fi'om  the  foot 
of   the    per})endicular. 

Tliis  property  of  the  sphere  is  used  n  the  discussion  of 
triangles   in    spherical   trigonometry. 


Il 


TRIGONOMETliY 


AND 


MENSURATION 


I 


« 


fflTPiODUCTION  TO  TRIGONOMETRY. 


LOGARITHMS. 

1.  TuE  Logarithm  of  a  number  is  the  exponent  of  the 
power  to  which  it  is  necessary  to  raise  a  fixed  number,  to 
produce   the   given    number. 

The  fixed  number  is  called  the  base  of  the  system.  Any 
positive  number,  excej^t  1,  may  be  taken  as  the  base  of  a 
system.       In    the   common    system,   the   base   is     10. 

2.  If    we    denote    any   positive    number  by    tt,      and    the 

corresponding    exponent    of    10,     by     a;,      we    shall    have    the 

exponential   equation, 

10'    =    n (1.) 

In  this  equation,  x  is,  by  definition,  the  logarithm  of  n, 
which   may   be   expressed   thus, 

X    =    log  n (2.) 

3.  From  the  definition  of  a  logarithm,  it  follows  that,  the 
logarithm  of  any  power  of  10  is  equal  to  the  exponent  of 
that  power  :    hence   the  formula, 

log  (10)"  =  p. (3.) 

If  a  number  is  an  exact  power  of  10,  its  logarithm  is 
a  ichole  member. 


4  INTRODUCTION. 

If  a  number  is  not  an  exact  power  of  10,  its  logarithm 
will  not  be  a  wbole  number,  but  will  be  made  up  of  an 
entire  part  plus  a  fractional  2^art,  which  is  generally  expres- 
sed decimally.  The  entire  part  of  a  logarithm  is  called  the 
c/iaracteristiCy   the    decimal  part,  is  called  the   mantissa. 

4.  If,  in  Equation  ( 3 ),  we  make  p  successively  equal 
to  0,  1,  2,  3,  &c.,  and  also  equal  to  —  0,  —  1,  —  2,  —  3, 
&c.,   we   may  form  the   following 


TABLE. 

log         1 

=. 

0 

log       10 

= 

1 

log      .1 

= 

-  1 

log     100 

z= 

2 

log    .01 

= 

-  2 

loor    1000 



3 

log  .001 

= 

-  3 

&c.,    &c.  &c.,    &c. 

If  a  number  lies  between  1  and  10,  its  logarithm  lies 
between  0  and  1,  that  is,  it  is  equal  to  0  ^jZws  a  deci- 
mal ;  if  a  number  Hes  between  10  and  100,  its  logarithm 
is  equal  to  1  plus  a  decimal ;  if  between  100  and  1000, 
its  logarithm  is  equal  to  2  plus  a  decimal ;  and  so  on : 
hence,   we   have  the   following 

EUL  E. 

The  characteristic  of  the  logarithm  of  an  entire  number  is 
positive,  and  numerically  1  less  than  the  number  of  places 
of  figures  in   the  given   number. 

If  a  decimal  fraction  lies  between  .1  and  1,  its  loga 
rithm  lies  between  —  1  and  0,  that  is,  it  is  equal  to  —  1 
plus  a  decimal  ;  if  a  number  lies  between  .01  and  .1,  its 
logarithm  is  equal  to  —  2,  plus  a  decimal ;  if  between  .001 
and  .01,  its  logarithm  is  equal  to  —  3,  plus  a  decimal ; 
and   80   on  :    hence,   the   following 


TRIGONOMETRY.  5 

KULB. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction 
is  negative,  and  numerically  1  greater  than  the  number 
of    O's    that  immediately  follow  the  decimal  point. 

The  characteristic  alone  is  negative,  the  mantissa  being 
always  positive.  This  fact  is  indicated  by  writing  the  neg- 
ative sign  over  the  characteristic :  thus,  2.371465,  is  equiv- 
alent  to     —  2  +  .371465. 

It  18  to  be  observed,  that  the  characteristic  cf  the  logarithm 
of  a  mixed  number  is  the  same  as  that  of  its  entire  part. 
Thus,  the  mixed  number  74,103,  lies  between  10  and  100; 
hence,  its  logarithm  lies  between  1  and  2,  as  does  the  logarithm 
of  74. 


GEIfERAL     PRIlSrCIPLES. 

5.     Let     m     and     n     denote    any  two    numbers,    and     a; 
and    y    their    logarithms.       We    shall    have,   from    the   defini 
tion   of  a  logarithm,   the  following   equations, 

10'  =    m (4.) 

10"  =    w (6.) 

Multiplying   (4)     and     (5),   member  by  member,   we   have, 

10'  +  "   =   mn; 
whence,   by   the   definition, 

X  -h  y   =   log  {mn) (6.) 

That    is,   the  logarithm    of  the  product    of    two    numbers    is 
equal  to   the  sum  of  the  logarithms  of  the  numbers. 


6  INTRODUCTION". 

6.  Dividing   ( 4 )    by    ( 5 ),    member  by   member,   w^e  have, 

■        ■  10-'  =   •") 

n 

whence,   by   the   definition, 

X  -  y   =   log  (^j (7.) 

That  is,  the  logarithm  of  a  quotient  is  equal  to  the  loga- 
rithm  of  the   dividend  diminished  by   that   of  the  divisor. 

7.  Raising  both   members  of    (4)     to   the   power  denoted 
by    jo,     we  have, 

whence,   by   the   definition, 

xp    =    log  m' (8.) 

That  is,  the  logarithm  of  any  power  of  a  number  is  eqital 
to  the  logarithm  of  the  number  multiplied  by  the  exponent 
of  the  power. 

8.  Extracting  the  root,  indicated  by    r,    of  both   members 
of    ( 4  )j      we   have, 

X 

whence,   by   the    definition, 

-    =    log  ym.     .     .     .     .     ( 9.) 


That  is,  the  logarithm  of  ay^y  root  of  a  number  is  equal 
to  the  logarithm  of  tJie  number  divided  by  the  index  of  tie 
root. 

The  preceding  principles  enable  us  to  abbreviate  the  oper 
atious  of  multiplication  and  division,  by  coitverting  them  Luto 
f.he   simpler   ones   of  addition    and   subtractiou. 


TRIGONOMETRY. 


TABLE    OF    LOGARITHMS. 

9,  A  Table  of  Logarithms,  is  a  table  containing  a  set 
of  numbers  and  their  logarithms,  so  arranged,  that  having 
given  any  one  of  the  numbers,  we  can  find  its  logarithm ; 
or,  having  the  logarithm,  Ave  can  find  the  corresponding 
number. 

In  the  table  appended,  the  complete  logarithm  is  given 
for  all  numbers  from  1  up  to  10,000.  For  other  number.-:, 
the  mantissas  alone  are  given;  the  charactei-istic  may  be  found 
by  one    of  the   rules   of  Art,  4. 

Before  explaining  the  use  of  the  table,  it  is  to  be  shown 
that  the  mantissa  of  the  logarithm  of  any  number  is  not 
changed  by  multi})lying  or  dividing  the  number  by  any  exact 
power   of    10. 

Let  n  represent  any  number  -whatever,  and  10^  any 
power  of  10,  p  being  any  whole  number,  either  positive 
or  negative.  Tlion,  in  accordance  with  the  principles  of  Arts. 
6    and    3,    we    shall    have, 

log  {n   X    lO'')    =    log  n  -h  log  lO''    =:   /)  +  log  n  ; 

but  p  is,  by  hypothesis,  a  whole  number  :  hence,  the  deci- 
mal part  of  the  log  {/I  X  lO'')  is  the  same  as  that  of  log  n  ; 
which   was   to  be  proved. 

Hence,  in  finding  the  mantissa  of  the  logarithm  of  a  num- 
ber, we  may  regard  the  number  as  a  decimal,  and  move  the 
dcchnal  point  to  the  right  or  left,  at  jileasure.  Thus,  the 
mantissa  of  the  logarithm  of  456:357,  is  the  same  as  that  of 
the  number  4563.57  ;  and  the  mantissa  of  the  logarithm  of 
2.00357,     is   the   same   as    that    of    2003.57. 


8  INTRODUCTION. 

MANNEB      OF     USES^G      THE     TABLE.  , 
1".     To  find  the  logarithm  of  a  number  less  than  100. 

10.  Look  on  the  first  page,  in  the  column  headed  *'N," 
for  the  given  number  ;  the  number  opposite  is  the  logarithm 
required.       Thus, 

log  67    =    1.826075. 

2".     To  find  the  logarithm   of   a    number    between     100     and 

10,000. 

11.  Find  the   characteristic  by  the  first  rule  of  Art.  4. 
To  find   the    mantissa,    look    in    the    column  headed   "  N,'* 

for  the  first  three  figures  of  the  number ;  then  pass  along 
a  horizontal  line  until  you  come  to  the  column  headed  with 
the  fourth  figure  of  the  number  ;  at  this  place  mil  be  found 
four  figures  of  the  mantissa,  to  Avhich,  two  other  figures, 
taken  from  the  colunm  headed  "  0,"  are  to  be  prefixed.  If 
the  figures  found  stand  opposite  a  row  of  six  figures,  in  the 
column  headed  "  0,"  the  first  two  of  this  row  are  the  onc3 
to  be  prefixed  ;  if  not,  ascend  the  column  till  a  row  of  six 
figures  is  found  ;  the  first  two,  of  this  row,  are  the  ones  to 
be   prefixed. 

If^  however,  in  passing  back  from  the  four  figures,  first 
found,  any  dots  are  passed,  the  two  figures  to  be  prefixed 
must  be  taken  fi-om  the  hne  immediately  below.  If  the 
figures  first  found  fall  at  a  place  where  dots  occur,  the  dots 
must  be  replaced  by  O's,  and  the  figures  to  be  prefixed  mist 
be   taken   from   the    Ime   below.       Thus, 

Log  8979  =  3.953228 
Log  3098  -  3.491081 
Log  2188    =    3.340047 


TRIGONOMETRY.  9 

3°.     To  fi)id  the  logarithm   of  a  number  greater  than  10,000. 

12.     Fiiid   the   characteristic   by   the   first  rule   of  Art.  4. 

To  fiucl  the  mantissa,  place  a  decimal  point  after  the  fonrtli 
figure  (Art.  9),  thus  converting  the  number  into  a  mixed 
number.  Find  the  mantissa  of  the  entire  part,  by  the  me- 
tliod  last  given.  Then  take  from  the  column  headed  "  D," 
the  corresponding^  tabular  difference,  and  multiply  this  by  the 
decimal  part  and  add  the  product  to  the  mantissa  just  found. 
The   result  will  be   the   required   mantissa. 

It  is  to  be  observed  that  when  the  decimal  part  of  the 
product  just  spoken  of  is  equal  to  or  exceeds  .5,  we  add 
1     to   the   entii'e  part,  otherwise   the  decimal  part  is  rejected. 

EXAMPLE. 

1.     To   find   the   logarithm   of   672887. 

The  characteristic  is  5.  Placing  a  decimal  point  after  the 
fourth  figure,  the  number  becomes  6728.87.  The  mantissa 
of  the  logarithm  of  6728  is  827886,  and  the  corresponding 
number  in  the  column  "D"  is  65.  Multiplying  65  by  .87, 
we  have  56.55  ;  or,  since  the  decimal  part  exceeds  .5,  57. 
We  add  57  to  the  mantissa  already  found,  giving  827943, 
and  we   finally   have, 

log  672887    =    5.827943. 

The   numbers  in   the    column    "D"   are   the   difierences  be- 
tween  the   logarithms  of  two   consecutive  whole   numbers,  and 
are  found  by  subtracting  the  nuniber    inder  the  heading  "  4  * 
fi"om  that   under   the   heading   "  5." 

In  the  example  last  given,  the  mantissa  of  the  logarithm 
of  6728  is  827886,  and  that  of  6729  is  827951,  and 
Uieir    drfference   is     65  ;      87   hundredths    of    this    difierence   is 


10  INTRODUCTION. 

57  :  hence,  the  mantissa  of  the  logarithm  of  6728.87  is  fomid 
by  addmg  57  to  827886.  The  principle  employed  is,  that 
the  differences  of  numbers  are  proportional  to  the  differences 
of  their   logarithms,   when   these   differences   are   small. 


4°.     To  find  the  logarithm   of  a   decimal. 

13.     Find  the   characteristic  by  the  second  rule   of  Art.  4. 

To   find  the   mantissa,  drop   the   decimal   point,  thus  reduc- 

inw   the   decimal   to   a  whole   number.       Fmd  the  mantissa   of 

the  logarithm    of   this    number,   and   it   will    be    the    mantissa 

required.       Thus, 

\o^       .0327     =    2.514548 
log  378.024    =    2.577520 


5°.     To  find  the  number  con'esponding  to  a  gicen  logarithm. 

14.  The  rule  is  the  reverse  of  those  just  given.  Look 
in  the  table  for  the  mantissa  of  the  given  logarithm.  If  it 
cannot  be  found,  take  out  the  next  less  mantissa,  and  also 
the  corresponding  number,  which  set  aside.  Find  tlie  differ- 
ence between  the  mantissa  taken  out  and  that  of  the  given 
logarithm ;  annex  as  many  O's  as  may  be  necessary,  and 
divide  this  result  by  the  corresponding  number  in  the  column 
"  D."  Annex  the  quotient  to  the  number  set  aside,  and  then 
point  off,  from  the  left  hand,  a  number  of  places  of  figures 
equal  to  the  characterististia  plus  1  :  the  result  will  be  the 
number  required.  K  the  characteristic  is  negative,  the  result 
wUl  be  a  pure  decimal,  and  the  number  of  O's  which  im- 
mediately  follow  the  decimal  point  will  be  one  less  than  the 
number   of  units  in  the   characteristic. 


TRIGONOMETRY.  H 


BZAMFLES. 

1.  Let  it  be  required  to  find  the  number  corresponding 
to    the    logarithm    5.2335G8. 

The  next  less  mantissa  in  the  table  is  233504  ;  the  cor- 
responding number  is  1712,  and  the  tabular  dilFerence  ie 
253. 

OPEKATION. 

Given   mantissa, 233568 

Next   less    mantissa,       •     •     •     233504     •     •     1T12 

253   )  6400000   (  25296 

.*.     The   required   mumber  is    171225.296. 

The  number  corresponding  to  the  logarithm  "2.233568  is 
.0171225. 

2.  What  is  the  number  corresponding  to  the  logarithm 
2.785407?  Ans.     .06101084. 

3.  What  is  the  number  corresponding  to  the  logaiithm 
1.846741  ?  Jins.     .702653. 


MULTIPLICATrOI^     BY     MEANS      OF     LOGARITITjrS. 

15.  From  the  principle  proved  in  Art.  5,  we  deduce  the 
following 

EXILE  . 

J^ind  the  logarithms  of  the  factors,  and  take  their  sum  , 
then  find  the  number  corresponding  to  the  resulting  logarithm, 
and  it  will  be  the  product  required. 


12  INTRODUCTION. 

BXAMPLES. 

1.  Multiply     23.14     by    5.062. 

OPERATION. 

log  23.14     .     .     •     1.364363 
log  5.062     .     •     •     0.704322 

2.068685     .♦.     117.1347,     product. 

2.  Find    the    continued    product     of     3.902,     597.16,     and 
0,0314728. 

OPEEATION. 

log  3.902     .     •     •     0.591287 

log         697.16     •     •     •     2.776091 
log  0.0314728     •     •     •     2.497936 

1.865314       .-.     73.3354,     product. 

Here,  the     2     cancels  the     +  2,     and  the     1     carried  from 
the   decimal  part  is   set   down. 

3.  Fmd  the  continued  product  of    3.586,     2.1046,     0.8372, 
and     0.0294.  Ans.     0.1857615. 


DIVISION      BY      MEANS      OF      LOGAEITHSIS. 

16.  From  the  principle  proved  in  Art.  6,  we  have  the 
following 

RULE. 

Find  the  logarithms  of  the  dividend  and  divisor^  and 
subtract  the  latter  from  the  former ;  then  find  the  number 
corresponding  to  the  resulting  logarithm^  and  it  will  be  the 
quotient  reqxdrcd. 


trigo:n  ometrt. 


13 


EXAMPLES. 

1.    Divide     24163     by     4567. 


log  24163 
log     4567 


OPEKATIOK. 

.     4.383151 
.     3.659631 

0.723520 


5.29078,     quotient. 


2      Divide     0.7438     by     12.9476. 

OPERATION. 


log     0.7438 
log  12.9476 


1.871456 
1.112189 

2.759267 


0.057447,    quotient. 


Here,  1  taken  from  1,  gives  2  for  a  result.  The 
subtraction,  as  in  this  case,  is  always  to  be  performed  in  the 
algebraic  sense. 


3.     Divide     37.149     by    523.76. 


Ans.     0.0709274. 


The  operation  of  division,  particularly  when  combined  with 
that  of  multiplication,  can  often  be  simplified  by  using  the 
principle   of 

THE     AEITHMETICAL      C05IPLEMENT. 

17.  The  AEmniETicAL  Complement  of  a  logarithm  is  the 
result  obtained  by  subtracting  it  from  10.  Thus,  8.130456 
is  the  arithmetical  complement  of  1.869544.  The  arithmetical 
complement  of  a  logarithm  may  be  wi-itten  out  6y  commenc- 
ing at    the  left  hand  and    subtracting    each  figure   from     9, 

18 


14 


INTRODUCTION. 


until  the  last  significant  figure  is  reached^  which  must  hi 
taken  from  10.  The  aritbmetical  complement  is  denoted  by 
the   symbol     (a.  c). 

Let  a  and  b  represent  any  two  logaritlims  whatever, 
and  a  —  h  their  difference.  Since  we  may  add  10  to, 
and  subtract  it  from,  a  —  b^  without  altering  its  value,  wo 
have, 

a  —  h    =    a  ■\-  {10  -  b)  -  \0.    .     .    .     (10.) 

But,  10  —  5  is,  by  definition,  the  arithmetical  complement 
of  b  :  hence.  Equation  (  10  )  shows  that  the  difference  be- 
tween two  logarithms  is  equal  to  the  firsts  plus  the  arith- 
m.etical  complement   of  the  second^    minus     10. 

Hence,  to  divide  one  number  by  another  by  means  of 
the   arithmetical   complement,   we  have  the   following 

RULE. 

Find  the  logarithm  of  the  dividend,  and  the  arithmetical 
complement  of  the  logarithm  of  the  divisor,  add  them  tog& 
ther,  and  diminish  the  sum.  by  10  ;  the  number  correspond 
ing   to   the  resulting  logarithm   will    be    the   quotient  required, 

EXAMPLES. 

1,     Divide     327.5     by     22.07. 

OPEKATION. 


log  32'7.5 
(a.  c.)  log  22.07 


2.5152U 
8.656198 

1.171409 


•.     14.839,     quotien\ 


2.     Divide     87 149     by     523.76. 


Ans.     0.0709273. 


TRIGONOMETRY, 


15 


3.    Multiply     358884     by     5G72,      and    divide    the   pioduct 
by     89721. 

OPERATION. 


log  358884 

log       5672 

(a.  c.)  log     89721 


5.554954 
3.753736 
5.047106 

4.355796 


22688,     result 


4.     Solve   the   proportion, 
397G    :     7952 


5903 


X. 


Applying  logarithms,  the  logarithm  of  the  4th  term,  is  equal 
to  the  sum  of  the  logarithms  of  the  2d  and  3d  terms,  minus  the 
logarithm  of  the  1st :  Or,  the  aritlimetical  comple7nent  of  the  1st 
term,  plus  the  logarithm  of  the  2d  term,  plus  the  logarithm  of  the 
3d  term,  minus  10,  is  equal  to  the  logarithm  of  the  Uh  term. 


(a.  c.)  log  3976 

log  7952 

log  5903 

log:  X 


OPERATION. 

.  .  G.400554: 
.  .  3.900476 
.     .     3.771073 


4.072103 


X  =  11806 


The  operation   of  subtracting  10,   is  performed  mentally. 


RAISING    OF   POWERS    BY    MEANS    OF   LOGARITHMS. 
18.    From  Article  7,  we  have  the  following 


RULE. 


Fi7id  the  logarithm  of  the  number,  and  multiply  it  hy  the 
exponent  of  the  poiuer ;  then  find  the  numher  corresponding  to 
the  resulting  logarithm,   and  Jt  tvill  he   the  poiver  required. 


16  INTRODUCTION. 

EXAMPLES. 

1.     Find  the   5th  power  of   9. 

OPERATION. 


los:  9     •     •     •     0.954243 

5 


4.771215      .-.      59049,    power. 
2.     Find  the   7th  power   of   8.  Ans.     2097152. 

EXTRACTLNa     EOOTS      BY     MEAISTS      OF     LOGAEITHMS. 

19.  From  the  principle  proved  in  Art.  8,  we  have  the 
following  I 

RULE. 

Find  the  logarithtn  of  the  number^  and  divide  it  by  the 
mdex  of  the  root  ;  then  find  the  number  corresponding  to 
the  resulting  logarithm^   and  it  will  be   the  root  required. 

EXAMPLES. 

1.  Find  the   cube   root   of   4096. 

The  logarithm  of  4096  is  3.612360,  and  one-third  of 
this  is  1.204120.  The  corresponding  number  is  16,  which 
16  the  root  sought. 

WJ\en  the  characteristic  is  negative  and  not  divisible  by 
the  index^  add  to  it  the  smallest  negative  number  that  will 
maJce  it  divisible,  and  then  prefix  the  same  number,  with  a 
plus   sign,   to   the   mantissa. 

2.  Find   the    4th   root   of   .00000081. 

The  logarithm  of  .00000081  is  7'.908485,  which  is  equal 
to    8  +  1.908485,     and   one-fourth    of  this    is    2.477121. 

The  number  corresponding  to  this  logarithm  is  0.'  : 
hence,    .03    is  the   root  required. 


PLANE    TRIGONOMETRL 


20  Plane  Tkigonometry  is  that  branch  of  Mathematics 
which   treats   of  the   solution   of  plane   triangles. 

In  every  plane  triangle  there  are  six  parts  :  three  sides 
and  three  angles.  When  three  of  these  parts  are  given,  one 
being  a  side,  the  remaining  parts  may  be  found  by  comput- 
ation. The  operation  of  finding  the  unknown  parts,  is  called 
the   solution  of  the  triangle. 


21.  A  plane  angle  is  measured  by  the  arc  of  a  circle 
included  between  its  sides,  the  centre  of  the  circle  being  at 
the   vertex,    and   its   radius  being   equal  to     1. 

Thus,  if  the  vertex  A  be  taken 
as  a  centre,  and  the  radius  AH  be 
equal  to  1,  the  intercepted  arc  BG 
will   measure   the   angle    A     (B.  III.,  P. 

xvn.,  s.). 

Let  ABCB  represent  a  circle  whose  radius  is  equal  to 
1,  and  AG^  BB^  two  diameters  per- 
pendicular to  each  other.  These  diar 
meters  divide  the  circumference  into 
four  equal  parts,  called  quadrants  ;  and 
because  each  of  the  angles  at  the  cen- 
tre is  a  right  angle,  it  follows  that  a 
right    angle    is    measured    by   a    quad- 


18 


PLANE     TRIGONOMETRY. 


ratit.  Au  acut6  angle  is  measured  by  an  arc  less  than  a 
quadrant^  and  an  obtuse  angle^  by  an  arc  greater  than  a 
quadrant. 


22.  In  Geometry,  the  unit  of  angular  measure  is  a  right 
angle  ;  so  in  Trigonometry,  the  "primary  unit  is  a  quctdranti 
which   is  the  measure   of  a  right   angle. 

For  convenience,  the  quadrant  is  divided  into  90  equal 
parts,  each  of  which  is  called  a  degree  ;  each  degree  into 
60  equal  parts,  called  tnijuites ;  and  each  minute  into  60 
equal  parts,  called  seconds.  Degrees,  minutes,  and  seconds, 
are  denoted  by  the  symbols  °,  ',  ".  Thus,  the  expression 
7**  22'  33",  is  read,  7  degrees,  22  minutes,  and  33  seco?ids. 
Fractional  parts  of  a    second   are   expressed   decimally. 

A  quadrant  contains  324,000  seconds,  and  an  arc  of  7° 
22'  33"  contains  26553  seconds ;  hence,  the  angle  measured 
by  the  latter  arc,  is  the  aVjoVoth  part  of  a  right  angle. 
In  like  manner,  any  angle  may  be  expressed  in  terms  of  a 
right   angle. 


23.    The  com2)lement  of  an  arc  is  the  difference  between 
that   arc   and  90°.      The   complement 
of   an    angle    is    the   difference    be- 
tween that  angle  and   a  right  angle. 

Thus,  ^B  is  the  complement  of 
A£J,  and  FJ3  is  the  complement 
of  AF.  In  like  manner^  FOB 
is  the  complement  of  A  OF,  and 
FOB    is  the  complement  of    A  OF. 

In  a  right-angled  triangle,  the 
acute   angles  are   complements   of  each   other. 


24.    The  supplemint  of  an  arc   is  the   difference  between 


PLANE      TRIGON  OMETUY. 


19 


that  arc  and  180°.  The  supplement  of  ati  angle  is  the  dif- 
ference  between   that   angle   and   two   right   angles. 

Thus,  EC  is  the  sujiplement  of  AE^  and  EG  the 
supplement  of  AF.  In  like  manner,  EOO  is  the  supple- 
ment  of   AOE^    and    FOC    the   supplement   of   A  OF. 

In  any  plane  triangle,  either  angle  is  the  supplement  of 
the   sum   of  the   other  two. 


25.  Instead  of  employing  the  arcs  themselves,  we  usually 
employ  certain  fwictions  of  the  arcs,  as  explained  below, 
A  function  of  a  quantity  is  something  which  depends  upon 
that   quantity  for   its   value. 

The  following  functions  are  the  only  ones  needed  for  solv- 
ing  triangles  : 

26.  The  sine  of  an  arc  is  the  distance  of  one  extremity 
of  the    arc   from   the   diameter,  through  the   other   extremity. 

Thus,  PM  is  the  sine  of 
AM^  and  F'M'  is  the  sine  of 
AM'. 

If  AM  is  equal  to  Jf' C, 
AM  and  AM'  will  be  supple- 
ments of  each  other  ;  and  be- 
cause MM'  is  parallel  to  A  C, 
FM  Adll  be  eqiial  to  F'M' 
(B.  I.,  P.  XXIII.)  :  hence,  tlie 
sine  of  an  arc  is  equal  to  the 
nine  of  its  supplement. 


27.  The  cosine  of  an  arc  is  the  sine  of  the  complement 
of  the    arc. 

Thus,  NM  is  the  cosine  of  AM^  and  iO/'  is  the 
cosine  of  AM'.  These  lines  are  respectively  equal  to  OP 
and    01 '. 


I 


20 


PLANE     TRIGONOMETRY. 


It  is  evident,  from  the  equal  triangles  of  tlie  figure,  that 
the  cosine  of  an  are  is  equal  to  the  cosine  of  its  supple 
ment. 


28.  The  ta?iffent  of  an  arc  is  the  perpendicular  to  the 
radius  at  one  extremity  of  the  arc,  limited  by  the  prolon- 
gation  of  the   diamater   through   the   other   extremity 

Thus,  AT  is  the  tangent  of 
the  arc  AM,  and  AT'"  is 
the   tangent   of  the   arc    AM'. 

If  AM  is  equal  to  J/' (7, 
A3I  and  AM'  will  be  supple- 
ments of  each  other.  But  AM'" 
and  A3I'  are  also  supplements 
of  each  other  :  hence,  the  arc 
AM  is  equal  to  the  arc  AM'", 
and     the     corresponding     angles, 

A  OM  and  A  031'",  are  also  equal.  The  right-angled  tri- 
angles AOT  and  AOT'",  have  a  common  base  AO,  and 
the  angles  at  the  base  equal  ;  consequently,  the  remainhig 
parts  are  respectively  equal:  hence,  AT  is  equal  to  AT'". 
But  AT  \s  the  tangent  of  A3r,  and  AT'"  is  the  tangent 
of  AM' :  hence,  the  tangent  of  an  arc  is  equal  to  the  ta^x- 
gent  of  its  supplement. 

It  is  to  be  observed  that  no  account  .s  taken  of  the  alge- 
braic signs  of  the  cosines  and  tangents,  the  numerical  valuee 
alone   being   referred   to. 

29  The  cotangent  of  an  arc  is  the  tangent  of  its  com- 
plement. 

Thus,  BT'  is  the  cotangent  of  the  arc  A3I,  and  BT" 
b  the   cotangent   of  the   ai'e    AM' . 

The  sine,  cosine,  tangent,  and  cotangent  of  an  arc,  a, 
are,   for  convenience,   U'ritten    sin  a,    cos  a,    tan  a,    and  cot  a. 


PLANE      TRIGONOMETRY. 


21 


These  functions  of  an  arc  have  been  defined  on  the  sup- 
position that  the  radius  of  the  arc  is  equal  to  1  ;  in  this 
case,  they  may  also  be  considered  as  functions  of  tlie  angle 
which   the   arc   measures. 

Thus,  7\T/,  i\"J/,  AT,  and  HT',  are  respectively  the 
sino,  cosine,  tangent,  and  cotangent  of  the  angle  A  031,  as 
TeU   as   of  the   arc    AM. 


30.  It  is  often  convenient  to  use  some  other  radius  than 
1  ;  in  sucli  case,  tlie  functions  of  the  ai'C,  to  tiio  radius  1, 
may  be  reduced  to   con-esponding  functions,  to   the  radius    li. 

Let  A  031  represent  any  angle, 
A3f  an  arc  described  from  0  as 
a  centre  "snth  the  radius  1,  Pil/" 
its  sine  ;  A'3I'  an  arc  described 
from  0  as  a  centre,  with  any  ra- 
radius  i?,  and  P'3I'  its  sine. 
Then,  because  0PM  and  OP'M' 
are   similar  triangles,   "we   shall  have, 

OM  :  PM  :  :   031'   :  P'M\    or,     1    :  P3I  :  :  2i  :  P'M'  \ 

whence. 


P3I  = 


P'3f' 
B 


and,      P'3r   =  P31  X  B; 


and  similarly   for   each   of  the   other   functions. 

That  is,  any  function  of  an  arc  whose  radius  is  1,  is 
!qual  to  the  corresponding  function  of  an  arc  whose  radius 
is  p.  divided  hy  that  radius.  Also,  a7iy  function  of  an 
arc  lohose  radius  is  B,  is  equal  to  the  corresponding  funo- 
tion  of  an  arc  whose  radius  is  1,  midtiplied  by  the  ra- 
dius   B. 

By  making  these  changes  in  any  formula,  the  formula  will 
be   rendered  homogeneous. 


22  PLANE     TRIGONOMETRY 

TABLE      OF      NATURAL      SINES. 

31.  A  Natubal  Sjlme,  Cosixe,  Tan-gent,  or  Coiangekt, 
is  the  sine,  cosine,  tangent,  or  cotangent  of  an  arc  whose 
radius  is   1. 

A  Table  op  Natural  Sines  is  a  table  by  means  of  which 
che  natural  sine,  cosine,  tangent,  or  3otangent  of  any  arc, 
may  be   found. 

Such  a  table  might  be  used  for  all  the  purposes  of  tri- 
gonometrical computation,  but  it  is  found  more  convenient  to 
employ  a  table  of  logarithmic  sines,  as  explained  in  the  next 
article. 

TABLE      OF     LOGARITHMIC      SINES. 

32.  A  Logarithmic  Sine,  Cosine,  Taxgent,  or  Cotan- 
gent is  the  logaritlim  of  the  sme,  cosine,  tangent,  or  cotan- 
gent  of  an    arc   whose  radius  is     10,000,000,000. 

A  Table  of  Logarithmic  Sines  is  a  table  from  which  the 
logarithmic  sine,  cosine,  tangent,  or  cotangent  of  any  arc  may 
be   found. 

The   logarithm   of  the   tabular   radius   is     10. 

Any  logarithmic  function  of  an  arc  may  be  found  by  mul- 
tiplying the  corresponding  natural  function  by  10,000,000,000 
(Art.  30),  and  then  taking  the  logarithm  of  the  result  ;  or 
more  simply,  by  taking  the  logarithm  of  the  corresponding 
*iatural  function,   and  then  adding    10    to   the   result   (Art.  5). 

33.  Li  the  table  appended,  the  logarithmic  functions  are 
given  for  every  minvte  from  0°  up  to  90°.  In  addition, 
their  rates  of  change  for  each  second^  are  given  in  the 
colunm  headed    "D," 

The  method  ol  computing  the  numbers  in  the  column 
headed  "D,"   will  be  understood  from  a  single  example.     The 


PLANE      TRIGONOMETRY.  23 

logarithmic  sines  of  27°  34',  and  of  27°  35',  are,  respect- 
ively, 9.665375  and  9.6G5617.  The  diiference  between  their 
mantissas  is  242  ;  this,  divided  by  60,  the  number  of  sec- 
onds in  one  minute,  gives  4,03,  Avhich  is  the  change  in  the 
mantissa  for     1",     between  the  Hniits     27°  34'     and     27°  35'. 

For  the  sine  and  cosine,  there  are  separate  cohunns  of 
iiiffercnces,  which  are  w'ritten  to  the  right  of  the  respective 
johimus  ;  but  for  the  tangent  and  cotangent,  there  is  but  a 
single  column  of  differences,  which  is  written  between  them. 
The  logarithm  of  the  tangent  increases,  just  as  fast  as  that 
of  the  cotangent  decreases,  and  the  reverse,  their  sum  being 
always  equal  to  20.  Tlie  reason  of  this  is,  that  the  product 
of  the  tangent  and  cotangent  is  always  equal  to  the  square 
of  the  radius  ;  hence,  the  sum  of  their  logarithms  must 
always  be   equal  to   twice   the  logarithm  of  the  radius,   or   20. 

The  angle  obtained  by  taking  the  degrees  from  the  top 
of  the  page,  and  the  minutes  from  any  hue  on  the  left  hand 
of  the  page,  is  the  complement  of  that  obtained  by  taking 
the  degrees  from  the  bottom  of  the  page,  and  the  mmutes 
from  the  same  line  on  the  right  hand  of  the  page.  But, 
by  definition,  the  cosine  and  the  cotangent  of  an  arc  are, 
respectively,  the  sine  and  the  tangent  of  the  complement  of 
that  arc  (Arts.  26  and  28)  :  hence,  the  columns  designated 
sine  and  tang^  at  the  top  of  the  page,  are  designated  cosine 
and  cotang   at   the   bottom. 

USE      OF      THE      TABLE. 

2b  find  the  logarithmic  functions   of   an    arc    which    is    ex- 
pressed in  degrees  and  minutes. 

34.  If  the  arc  is  less  than  45°,  lOok  for  the  degrees  at 
the  top  of  the  page,  and  for  the  minutes  in  the  left  hand 
■X)lunm  ;    then  follow  the   corresponding  horizontal  line  till  you 


24  PLANE     TRIGONOMETRY. 

come  to  the  column  designated  at  the  top  by  sine,  cosine^ 
tang,  or  cotang,  as  the  case  may  be ;  the  number  there 
foimd   is  the  logarithm  required.       Thus, 

log  sin  19°  55'     .     •     •     9.532312 
log  tan  19°  55'     .     •     •     9.559097 

If  the  angle  is  greater  than  45°,  look  for  the  degrees  at 
the  bottom  of  the  page,  and  for  the  minutes  in  the  right 
hand  column  ;  then  follow  the  corresponding  horizontal  line 
backwards  till  you  come  to  the  column  designated  at  the  bot- 
tom by  sine,  cosine,  tang,  or  cotang,  as  the  case  may  be  ; 
the   number   there   found   is   the   logarithm   required.       Thus, 

log  cos  52°  18'     .     .     •     9.786416 
log  tan  52°  18'     •     •     •  10.111884 


To  find  the  logarithmic  functions    of  an    arc  which    is    &a- 
pressed  in   degrees,   minutes,   and  seconds. 

35.  Find  the  logarithm  corresponding  to  the  degrees  and 
minutes  as  before  ;  then  multii)ly  the  corresponding  number 
taken  from  the  column,  headed  "D,"  by  the  number  of  sec- 
onds, and  add  the  product  to  the  preceding  result,  for  the 
sine  or  tangent,  and  subtract  it  therefrom  for  the  cosine  or 
cotangent. 

EXAMPLES. 

1.     Find   the   logarithmic   sine   of    40°  26'  28". 

OPERATION. 

log  sm  40°  26' 9.811952 

Tabular   difference     2.47 
No.  of  seconds  28 

Product      •     •     .     69.16     to  be  added     •     •  69 

log  sin  40°  26'  28" 9.812021 


PLANE     TRIGONOMETRY.  25 

The  same  rule  is  followed   for  decimal  partSj  as  in  Art.  12. 

2.  ,  Find   the   logarithmic   cosine   of    53°  40'  40". 

OPEKATION. 

log  cos  53°  40' 9.772675 

Tabular  difference     2.86 

No.   of  seconds  40 

Product    •     •     •     114.40      to  be  subtracted  114 

log  cos  53°  40'  40"        9.772561 

If   the    arc    is    greater    than      90°,      we   find    the   required 
ftiuction   of  its   supplement  (Arts.  26    and    28). 

3.  Find  the  logarithmic  tangent   of    118°  18'  25". 

OPEKATION. 

180° 

Given   arc 118°  18'  25" 

Supplement        61°  41'  35" 

log  tan  61°  41' 10.268556 

Tabular  difference     5.04 
No.  of  seconds  35 

Product    .     .     •     176.40      to  be  added      •     176 

log  tan  118°   18'  25" 10.268732 


4.    Find   the  logarithmic   sine   of    32°  18'  35". 

A?is.     9.727945. 

6,     Find   the   logarithmic   cosine   of    95°  18'  24". 

Ans.     8.966080. 

6.     Find  the   logarithmic   cotangent   of    125°  23'  .'iO". 

Ans.      9.851619. 


26  PLANE     TRIGONOMETRY. 

To  find  the   arc   corresponding   to    any  logarithmic  function, 

36.  This  is  done  by  reversing  the  preceding  rule  : 
Look  in  the  proper  column  of  the  taLlo  for  the  given  log- 
ai'ithm  ;  if  it  is  found  there,  the  degrees  are  to  be  taken 
from  the  top  or  bottom,  and  the  minutes  from  the  left  or 
right  hand  column,  as  the  case  may  be.  If  the  given  log- 
arithm is  not  found  in  the  table,  then  find  the  next  less 
logarithm,  and  take  from  the  table  the  corresponding  degrees 
and  minutes,  and  set  them  aside.  Subtract  the  logarithm 
found  in  the  table,  from  the  given  logarithm,  and  divide  the 
remainder  by  the  corresponding  tabular  difference.  The  quo 
tient  will  be  seconds,  which  must  be  added  to  the  degrees 
and  minutes  set  aside,  in  the  case  of  a  sine  or  tangent,  and 
subtracted,  in   the   case   of  a   cosine   or   a   cotangent. 

EX  AMP  LES. 

1.  Find     the     arc     corresponding     to     the    logarithmic 
sine  9.422248. 

OPERATION'. 

Given   logarithm       •     •     •     9.422248 

Next  less  in  table  •     •     •     9.421857       •     •     •     15°  19' 

Tabular  difference         7.68)  391.00(51",   to  be  added. 

Hence,   the   required   arc   is     15°  19'  51". 

2.  Find     the     arc     corresponding     to     the     logarithmic 
cosine  9.427485. 

OPKRATIOX. 

Given   los^arithm       •     •     •     9.427485 

Next  less  in   table       •     •     9.427354      ...     74°  29'. 

Tabular   difference         7.58  )  131.00  (  H  ,     to   be   subt 

Hence,  the   required   arc   is     74°  28'  43". 


PLANE      T  R I G  O  N  0  M  E  T II Y . 


27 


3.  Find     the     arc  corresponding     to     the     logarithmic 
Mue    9.880054.  A>is.     49=  20'  50". 

4.  Find     the      arc  corresponding     to      the     logaritlimic 
cotangent    10.008688.  A?is.     44°  25'  37". 

5.  Find     the    arc  corresponding     to     the    logai-ithmic 
cosine    9.944599.  Ans.     28°  19'  45". 


SOLUTION      OF     RIGHT-ANGLED      TRIANGLES. 

37.  In  Avhat  folloAvs,  we  shall  designate  the  throe  angles 
of  every  triangle,  by  the  capital  letters  A,  B,  and  C.  A 
denoting  the  right  angle  ;  and  the  sides  lying  opi)Osite  the 
angles,  by  the  corresponding  small  letters  a,  5,  and  c. 
Since  the  order  in  which  these  letters  are  placed  may  be 
changed,  it  follows  that  whatever  is  proved  with  the  letters 
placed  in  any  given  order,  -vntII  be  equally  true  when  the 
letters   are   correspondingly  placed   in    any  other   order. 

Let  CAH  represent  any  triangle, 
right-angled  at  A.  With  C  as  a 
centre,  and  a  radius  CD,  equal  to  1, 
describe  the  arc  DG,  and  draw  GF 
and  DU  perpendicular  to  CA  :  then 
will  I^G  be  the  sine  of  the  angle  C,  CF  will  be  its 
cosine,   and    DF    its   tangent. 

Since  the  three  triangles  CFG,  CDE,  and  CAB  are 
dmilai  (B.  Vs  .,  P.  XVItl.),  we  may  write  the  proper 
tions. 


F  \) 


CB  :  AB 

:  CG  :  FG, 

or, 

a 

:    c    :  : 

1 

:     sin     C 

CB  :  CA  : 

:  CG  :  CF, 

or, 

a 

:    J    :  : 

1 

cos     C 

CA  :  AB  : 

:  CD  :  DE, 

or, 

I 

:     c    :  : 

1 

tan     C 

sin  6  =    — ,     •     ' 
a 

•     (4.) 

-  .*.  - 

cos  (7   =  —  ,    • 
a 

•     •     (5.) 

tan  G  —  -J-  i    ' 

.     .     (6.) 

28  PLANE     TRIGONOMETRY, 

hence,   we   have    (B.  II.,    P.  I.), 

c  =  a  sin  (7     •     •     •     (1.) 

6  =  a  cosC     .     .     .     (2.) 

c  —  b  UnC    •     •     •     (3.) 

Translating  these  formulas  into  ordinary  language,  we  have 
the  following 

PRINCIPLES. 

1.  The  perpendicular  of  any  right-angled  triangle  is  equat 
to  the  hypothenuse  into  the  si?ie  of  the  angle  at  the  base. 

2.  The  base  is  equal  to  the  hypothenuse   into  the  cosine 
of  the  angle  at   the  base. 

3.  The  perpendicidar  is  equal    to   the  base  into   the  tan- 
gent of  the  angle  at  the  base. 

4.  The  si7ie  of  the  angle  at  the  base  is  equal  to  the 
perpendicular   divided  by   the  hypothenuse. 

5.  The  cosiiie  of  the  angle  at  the  base  is  equal  to  the 
base  divided  by  the  hypothenuse. 

6.  The  tangent  of  the  angle  at  the  base  is  equal  to  the 
perp)endicidar  divided  by  the  base. 

Either  side  about  the  right  angle  may  be  regarded  as  the 
base;  in  which  case,  the  other  is  to  be  regarded  as  the 
perpendicular.  We  see,  then,  that  the  above  prmciples  are 
sufficient  for  the  solution  of  every  case  of  right-angled  tri- 
angles. When  the  table  of  logarithmic  sines  is  used,  in  the 
solution,  Formulas  (1)  to  (6)  must  be  made  homogeneous, 
by  substituting    for    sin  (7,     cos  C,     and    tan  (7,     respectively, 


PLANE     TRIGONOMETRY.  29 

5in  C  cos  (7  tan  G  _ 

— D~  >        — p —  »        ^^^  T>     »  -^      being     equal     to 

10,000,000,000,     as   explained  in  Ai't.  30. 

Making   these   changes,   and    reducing,   we   have, 

a  sin  (7  .  Re  ^      ^ 

c    =    — 5^ —     •    •    •     (7.)  sm  (7    =    —    •    •    •    (10) 

b    =    ^  ^^"^    ...     (8.)  cos(7    =    :^^    .    .        (11.) 


c    = 


a  sin  C 

•     (^.) 

sm  C  =  —  .  . 
a 

n      ■   ■ 

a  cos  C 

.     (8.) 

cos  C  =  — ■  .  . 
a 

b  tan  C 

•     .     (9-) 

tan  C    =    —r-     •     . 

0 

(12.) 


In  ajjplying  logarithms  to  these  formulas,  remember,  that 
the  sum  of  the  logarithms  of  the  two  terms  which  multiply 
together,  is  equal  to  the  sum  of  the  logarithms  of  the  other 
two  terms,  and  that  the  required  term  comes  last  in  the 
operation.  Also,  that  the  logarithm  of  E  is  10,  and  the 
arithmetical   complement   of  it,   is  0. 

There  are  four  cases. 

CASE   I. 

Given   the   hyjjothenuse    and    one    of  Lite   acute  angles,   to  find 

the  remaining  ])arts. 

38.  The  other  acute  angle  may  be  found  by  subtracting 
the   given   one   from   90°    (Art.  23). 

The  sides  about   the  right  angle  may 
be   found   by   Formulas    ( 7 )    and    ( 8). 


EXAMPLES.  U         b  A 

1.     Given    a    =    749,    and    C  =   47'    03'    10";     required 
By   c,   and   i. 

OPERATIOJSr. 

j5  =  90°  -  47°  03'  10"  =  42°  56'  50". 

Applying  logarithms   to   formula   ( 7 ),   we  have, 

19 


30  PLANE    TRIGONOMETRY. 

log   a  +   log   sin    C  —   10  =  log   c\ 

log  a  (749)     ....    2.874483 

log  sin   C    (47°  03'  10")     .     9.864501 

log  c 2.738983     .-.    c  =  548.255. 

Applj^ng  logarithms  to  Formula   (8),  we  have, 
log  a  +  log  cos    C  —   10  =  log  h ; 

log  a    .         (749)     ....     2.874481 
log  cos  C    (47°  03'  10")     .     9.833354 

log  5 2.707835     .-.    I  =  510.31. 

Ans.   B    =    42°    56'    50",    b   =   510.31,    and    c  =  548.255. 

2.     Given     a   =   439,    and    B    =    27°    38'    50",    to    find 
C,   c,   and  h. 

OPERATION". 

C  =  90°  -  27°   38'   50"  =i  62°  21'   10" ; 

log  a  (439)     ....    2.642465 

log  sin  C    (62°  21'  10")     .    9.947346 

log  c 2.589811     .-.     c  =  388.875. 

log  a  (439)     ....     2.642465 

log  cos  C    (62°  21'  10")     .     9.666543 

log  6 2.309008     .-.     h  =  203.708. 

Ans.    C  =    62°   21'   10",    h    =    203.708,  and  c   =    388.875. 

3      Given    a   =    125.7    yds.,    and   B   =    75°    12',    to    find 
the   other  parts. 

A71S.   C  =   14°   48',   I)   =   121.53   yds.,   and  c  =  32.11  yds. 


PLANE      TRIGONOMETRY.  31 

CASE      II. 

Given   one  of   the  sides    about    the    right  angle   and  one  of 
the  acute  angles,   to  find  the  remaining  parts. 

39.  The  other  acute  angle  may  be  found  by  subtracting 
the   given   one   from    90°. 

The  hypothenuse  may  be  found  by  Formula  (7),  and 
the   unknoAvn   side   about   the   right  angle,  by  Formula    (8). 

EXAMPLES. 

1.  Given  c  =  56.293,  and  C  =  54°  27'  39",  to  find  B, 
a,     and    b. 

OPERATION. 

J5    =    90°  —    54°  27'  39"  =  35°  32'  21". 

Applying  logarithms   to   Formula    (7),   we   have, 

log  c  +  10  —  log  sin   0  =  log  a; 

but,  10  —  log  sin  C  =  (a.  c.)  of  log  sin  C;  whence, 

log  c  (56.293)     .     .     .     1.750454 

(a.  c.)  log  sin  C     (54°  27'  39")     .     0.089527 

log  a 1.839981     .-.    a  =  G9.18. 

Applying  logarithms  to  Formula   (8),   we   have, 

log  a  +  log  cos  6'  —  10  =  log  b ; 

log  a  (69.18)       .    .    .    1.839981 

log  cos  C    (54°  27'  39")     .     9.764370 

log  ^ 1.604351     .-.    b  =  40.2114. 

Ans.  B   =   35°    32'    21",    a   =    69.18,    and   b   =   40.2114. 


32  PLANE     TRIGONOMETRY. 

2.  Given     c  =  358,     and     B  =  28°  47',     to   find    (7,     ff. 
and    6 

OPERATION. 

(7    _    90°  _  28°  47'    =    61°  13'. 
We  have,   as  before, 

log  c  +  10  —  log  sin  C  =  log  a ; 

log  c  (358)       .    .    .    2.553883 

(a.  c.)  log  sin  G    (Gl°  13')     .    .     0.057274: 

\oz  a 2.611157    .'.    a  =  408.466; 

Also,  log  a  +  log  cos   C  -  10  =  log   I; 

log  a  (408.4GG)    •     .     2.611157 

log  cos  G     (61°    13')    •     .     9.682595 

W  6 2.293752     .'.     h  =   196.670. 

Ans.     G  =  61°  13',      a  =   408.466,      and     b  —  196.676. 

3.  Given     h  =  152.67  yds.,      and     G   =    50°  18'  32",     to 
find   the   other  parts. 

Ans.     B  =   39°  41'  28",     c  =   183.95,    and    a  =  239.05. 

4.  Given     c  =  379.628,      and     C  =   39°  26'  16",      to  find 
Bf     a,      and     b. 

Ans.     B  =  50°  33'  44",     a  =  597.613,     and     b  —  461.55. 

CASE      m. 

Given    the    two   sides    about    the  rig\t    angle,  to  jind  the    re 

maining  parts. 

40.     The  angle   at    the    base    may  be    found    by  Formula 
(12),    and   the   sohition   may  be   completed   as   in    Case   IL 


PLANE     TRIGONOMETRY.  33 

EXAMPLES. 

1.  Given     h  =  26,    and    c  =  15,    to  find    C,   B,  and  a. 

OPERATION. 

Applj-ing  logaritlims  to  Formula   (12),  we  have, 

log  c  +  10  —  log  b  =  log  tan  C; 

log  c    (15)     ....    1.176091 
(a.  c.)  log  I)    (20)     ....    8.585027 

log  tan  C     ...     9.761118    .-.  C  =  29°  68'  54"; 

B  =  dO°  —  C^  60°  01'  06  ■. 
A3  in   Case  II.,   log   c  +  10  -  log  sin   0  =  log  a; 

log  c        .     .     (15)     .     .     ]  176091 
(a.  c.)  log  sm  C     (29°  58'  54")     0.301271 

log  a 1.477362       .-.     a  =  30.017. 

A?is.     C  =  29°  58'  54",     B  =  60°  01'  06",     and     a  -  30.017. 

2.  Given     b  =   1052  yds.,    and    c  =  347.21  yds.,    to   find 
j5,     C,      and      a. 

B  =  Yl°  44'  05",    C  =  18°  15'  55",   and   a  =  1107.82  yds 

3.  Given     b  =  122.416,     and      c  =   118.297,     to   find    B, 
C\    and    a. 

B  =  45°  58'  50",     C  =  44°  1'  10",     and     a  =  170.226 

4.  Given      b  =  103,       and       c  =  101,      to   fird     J5,     0 
and    a. 

J9  =  46°  33'  42",     C  =   44°  26'  18",     and     a  =  144.250. 


34  PLANE     TRIGONOMETRY. 


CASE      IV. 

Given  the  hypothenuse  and  either  side  about  the  right  angle, 
to  find  the  remaining  parts. 

41.  The  angle  at  the  base  may  be  found  by  one  oF 
Formulas  (10)  and  (11),  and  the  remaining  side  may  then 
be  found   by   one   of  Formulas   ( 7 )    and   ( 8 ). 

EXAMPLES. 

1.  Given  a  =  2391.76,  and  h  =  385.7,  to  find  (7, 
B,    and    c 

OPERATIOIf. 

Applying  logarithms   to  Formula   (11),   we  have, 

log  5  +  10  —  log  a  =  log  cos   C; 

log  b     (385.1)    •     •     •     2.586250 
(a.  c.)  log  a     (2391.76)     •     •     6.621282 

log  cos  (7      •     •     •     9.207532    .'.    C  =  80°  43' 11"; 

^  =   90°  —  80°  43'  11"   =    9°  16'   19". 

From   Formula   (  7 ),   we   have, 

log  a  +  log  sin   C  —  10  =  log  c; 

log  a  (2391.76)       •     3.378718 

iOg  sin  C     (80°  43'  11")     9.994278 

log  c 3.372996       .'.     c  =  23fl0.4C. 

Am.     ^  =  9*  16'  49",      C  =  80°  43'  11",      and      c  =  2360.45. 


PLANE    TRIGONOMETRY.  35 

# 

2.     Giveu   a  =   127.174  yds.,  and  c  =   125.7  yds.,   to  Qud 
C    B,   and   b. 


OPEKATION". 

From  Formula   (10),   we  have, 

log  c  +  10  —  log  a  =  log  sin   C; 

log  c     (125.7)     .    .    .    2.099335 
(a.  c.)  log  a     (127.174)      .    .     7.895602 

log  sin  C     ...     9.994937    .*.     C  =    81°   16'   6"; 
^  =  90°  -  81°  16'  6"  =  8°  43'  54". 
From   Formula    (8),   we   have, 

log  a  +  log  cos   (7  —  10  =  log  b; 

log  a  (127.174)        •    2.104398 

log  cosC     (81°   16'  6")      .     9.181292 

log  6 1.285690        .-.     b   =    19.3. 

A?is.     £  =   8°  43'  54",      G  =   81°  16'  6",     and   b  =   19.3  yds. 

3.  Given   a  =  100,     and    b  =  60,     to   find    -B,     C,    and    & 
Ans.     B  =  36°  52'  11",     C  =  53°  7'  49",     and   c  =  80. 

4.  Given    a   =   19.209,       and   c   =   15,       to    find      jB,     C\ 
and     b. 

A71S.     B  z=   38°  39    30"        C  =   5r  20'  30",     b  =   12. 


86 


PLANE      TRIGONOMETRY. 


SOLUTION      OF      OBLIQUE-ANGLED      TRIANGLES- 

42.     In    the   solution   of  oLlique-anglecl   triangles,  four  capes 
may  arise.       We   shall    discuss   these    cases   in   order. 


CASE      I. 

Given   one  side   and    two   angles^    to   determine   tJie  remaini/ig 

parts. 

43.  Let  ABC  represent  any 
oblique-angled  triangle.  From  the 
vertex  (7,  draw  CD  perpendicular 
to  the  base,  forming  two  right- 
angled  triangles  A  CD  and  B  CD. 
Assume  the  notation   of  the  figure. 

From    Formula    (  1  ),    we   have, 

CD    =    b  sin  A^  and         CD    =    a  sin  B  ; 

Equating   tliese   two    values,    we   have, 

J  sin  ^    =    a  sin  ^  ; 
whence    (B.  IL,   P.  II.), 

a     :      b      :  :      sin  A     :     Bm  B.       .     .     ( 13.) 

Since  a  and  b  are  any  two  sides,  and  A  and  B  the 
angles  lying  opposite  to  them,  we  have  the  following  princi- 
ple : 

The  sides  of  a  plane  triangle  are  proportioiial  to  tin 
sines   of  their  opposite  angles. 

It  is  to  be  observed  that  Formula  (13)  is  true  for  any 
vrtlue  of  the  radius.  Hence,  to  solve  a  triangle,  when  a  side 
aiid  two  angles  are  given : 


PLANE     TRIGONOMETRY.  37 

First  find  the  third  angle,  by  subtracting  the  sum  of  the 
given  angles  from  180°  ;  then  find  each  of  the  required  sides 
by  means   of  the  principle  just  demonstrated. 

EXAMPLES. 

1.     Given     B  =  58°  07',     C  =  22°  37',     and     a  =   408,  to 
find    A,      b,     and     c. 

OPERATION. 

-B 58°  07' 

C 22°  37' 

A      .     .     .     180°  —   80°  44'    =    99°  16'. 

To  find     by     write   the   proportion, 

sin  ^4     :     sin  J5      :  :      a      :     b  ; 

that  is,  the  sine  of  the  angle  opposite  the  given  stde,  is  to 
the  sine  of  the  angle  opposite  the  required  side,  us  the  given 
side  is   to   the  required  side. 

Applying  logarithms,   we   have    (Ex.   4,   P.    15), 

(a.  c.)  log  sin  A  +  log  sin  ^  +  log  a  —  10  =  log  b , 

(a.  c.)  log  sin  A  (99°  IC)     .    .    .  0.005705 

log  sin  B  (58°  07')     .    .    .  9.928972 

log  a  .    .  (408)      ....  2.610660 

log   J 2.545337    .*.     J  =  351.024. 

In   like  manner,   sin   J    :    sin  (7   :  :    a    :    c; 

and,      (a.  c.)  sin  A  +  log  sin   C  +  log  a  —  10  =  log  c. 

(a.  c.)  log  sin  A     (99°  16')     .     .    .  0.005705 

log  sin  C    (22°  37')     .    .    .  9.584968 

log  a  .    .     (408)      ....  2.610660 

log  c 2.201333    .-.     c  =  158.976. 

Ans.     A    =    99°     16',     b   =   351.024,     and     c   =    158.976, 


38 


PLANE     TRIGONOMETRY. 


2.  Given     A  =   38°  25',      Ji  =  57°  42',      and      c  =  400, 
to  find    C,      a,      and     6. 

^/is.     (7  =   83°  53',      a  =   249.974,      b   =   340.04. 

3.  Given       A   =    15°  19'  51",        C   =   72°  44'  05",       and 
c  =  250.4   yds,   to   find     i?,      a,      and     b. 

Ans.     B  =   91°  56'  04",     a  =  69.328  yds.,     b  =  262.066  yda. 

4.  Given  i?  =  51°  15'  35",  C  =  37°  21'  25",  and 
a  =  305.296  ft.,   to   find    A^      b^      and     c. 

Ans.     A   =   91°  23',     b  =   238.1978    ft.,     c  =  185.3  ft. 

CASE     n. 

Given  two  sides  and  an  angle  opposite  one  of  them^  to  find 

the  remaining  parts, 

44,  The  solution,  in  this  case,  is  commenced  by  finding 
&  second  angle  by  means  of  Formula  (13),  after  which  we 
may  proceed  as  in  Case  I.  ;  or,  the  solution  may  be  com- 
pleted by   a   continued   application   of   Formula   (13). 


EXAMPLES. 

1.     Given     A  =  22°  37',      b  =  216,      and     a  -.-  117,     to 
find    jB,     Cy      and      c. 

From  Formula    (13),   we   have,  ♦ 

a     :      b     :  :      sin  ^     :     sin  ^  ; 

that  is,  the  side  opposite  the  given  angle,  is  to  the  side  op- 
posite the  required  angle,  as  the  sine  of  the  given  angle  is 
to  the  sine  of  the  required  angle. 


PLANE    TRIGONOMETRY.  39 

Whence,  by  the  application   of  logarithms, 

(a.  c.)  log  a  +  log  d  +  log  sin  ^  —  10  =  log  sin  B ; 

(a.  c.)  log  a       .    .     (117)     .    .  7.931814 

log  5       .    .     (216)     .    .  2.334454 

log  sin  A  (22°  37')     .    .  9.584968 

log  sin  ^      ....  9.851236  .'.  B  =    45°  13'  55", 

and    B'  =  134°  46'  05". 

Hence,  we  find  two  values  of  i>,  which  are  supplements  of 
each  other,  because  the  sine  of  any  angle  is  equal  to  the 
sine  of  its  supplement.  This  would  seem  to  indicate  that 
the  problem  admits  of  two  solutions.  It  now  remains  to 
determine  under  what  conditions  there  will  be  two  solutions, 
one  solution,   or  no   solution. 

There  may  be  two  cases  :    the  given  angle  may  be  acvte, 
or  it   may   be   obtuse. 

First  Case.  Let  ABC  re- 
present the  triangle,  in  which  the 
angle    A,     and  the  sides    a    and 

b    are   given.       Fi-om    C    let  faU  "~- •'' 

a    perpendicular   upon    AB,    pro- 
longed  if  necessary,  and   denote   its  length  by    p.      We   shaU 
have,  from  Formula   ( 1 ),   Art.  37, 

p   =    b  sin  A  I 

from  which   the   value   of   p    may  be    computed. 

If  a  is  intermediate  in  value  between  p  and  b,  there 
wiU  be  tico  solutions.  For,  if  with  (7  as  a  centre,  and  a 
as  a  radius,  an  arc  be  described,  it  will  cut  the  line  AM 
in  two  pomts,  B  and  B',  each  of  which  being  joined  with 
(7,  will  give  a  triangle  which  wiU  conform  to  the  conditions 
of  the   problem. 


•10 


PLANE     TRIGONOMETRY. 


In   tliis  case,  the  angles    D'    and    -Z?,    of  the  two  triangles 
AB'C    and    AUC,    will  be   supplements   of  each  other. 

C 

If     a    =  py      there  will   be   but 

one    solution.       For,    in    this    case, 
the   arc   will    be    tangent    to     AB^ 
the    two    points     B    and     B'     will 
unite,  and  there  will  be  but  a  single  triangle  formed. 
In   this   case,   the   angle    ABC    will    be   equal    to   90°. 

If  a  is  greater  than  both  p 
and  J,  there  w^ill  also  be  but  one 
solution.  For,  although  the  arc 
cuts  AB  in  two  points,  and  con- 
sequently gives  two  triangles,  only- 
one  of  them  conforms  to  the  con- 
ditions  of  the    problem. 

In   this   case,  the  angle    ABC    will   be   less   than    A^    and 
consequently  acute. 


If  a  <.  p^  there  will  be  no 
solution.  For,  the  arc  can  neither 
cut    ABy     nor   be   tangent   to   it. 


Second    Case.     When    the    given    angle    A     is   obtuse,   the 
angle     ABC     will    be    acute  ;    the 
side     a      will    be    greater    than     5, 
and    there    will    be    but    one    solu-  / 

tion.  / 

4. 

\.n   the  example  under  considera-  ""--»._ 

tion,   there    are    two    solutions,    the 

first   corresponding   to     ^  =  45°  13'  55",     and   the  second    to 
B'  =  134°  46'  05". 


PLAXE    TRIGOXOMETRY.  41 

In    tlie   first  case,  we  have, 

A 22°  37' 

B 45°  13'  55^' 

C 180°  -  67°  50'  55"    =    112°  09'  05". 

To   find  c,   we  have, 

sin  B  :   sin  C  :   :   b  :   c;    and 

(a.  c.)  sin  B  +  log  sin  C  +  log  J  —  10  =  log  c; 

(a.  c.)  log  sin  B     (  45°  13'  55")    .  0.1487G4 

log  sin  C     (112°  09'  05")    .  9.96G700 

log  5     ...     (21G)     .    .    .  2.334454 

log   c 2.440918    .-.    c  =  281.785. 

Ans.    ^  =  45°  13'  55",      C  =  112°  09'  05",     and  c  =  281.785. 

In   the   second   case,   we  have, 

A 22°  37' 

B' 134°  46'  05" 

G 180°  -  157°  23'  05"    =    22°  36'  55"; 

and   as  before, 

(a.  c.)  log  sin  B'   (134°  46'  05")  .  0.148764 

log  sin  0    (  22°  36'  55")  .  9.584943 

log  5     .    .    .     (216)     .    .  .  2.334454 

log  c  .......  .  2.068161    .-.    c  =  116.993. 

Ans.    B'  =  134°  46'  05",     C  =  22°  36'  55",     and  c  =  116.993. 

2.    Given   A    =    32°,     a   =    40,     and     5    =    50,     to    find 
B,    C,    and    c. 

B  =    41°  28'  59",     C  =  106°  31'  01",    c  =  72.368 
Ans.  ■{ 

B  =  138°  31'  01",     C=      9°  28'  59'',     c  =  12.436. 


42  PLANE     TRIGONOMETRY. 

3.     Given      A    =    18°  52'  13",         a    =    2V.465  yds.,      and 
b  =   13.189  yds.,    to   find    i?,     C,      and      c. 

Ans.     a  =   B°  56'  05",     C  =   152°  11'  42",     c  =  39.011  yds. 


4.     Given      A    =    32°  15'  26",        b    =    1Y6.21  ft.,        and 
a  =  94.047  ft.,      to  find      i?,      (7,      and     c. 

Ans.     B  =  90°,       (7  =  57°  44'  34",      c  =  149.014  ft. 


CASE     in. 

Given    two    sides    and    their    included  angle^    to  find  the   ro- 

maining  parts. 

45.  Let  ABC  represent  any 
plane  triangle,  AB  and  AG  any 
tAVO  sides,  and  A  their  included 
angle.        With     ^      as    a    centre, 

and    AC^     the  shorter  of  the  two  c'— --'F    H 

sides,  as  a  radius,  describe  a  semi- 
circle  meeting     AB     in     7,      and    the    prolongation    of    AB 
in    E.       Draw     CI     and    EC,     and  through     I     draw    ZS" 

parallel   to    EC. 

Since  the  angle  C^^  is  exterior  to  the  triangle  CBA, 
we  have    (B.   L,   P.  XXV.,   C.   6), 

CAE  =  C  +  B. 

But  the  angle   CIA   is  half  the  angle   CAE; 
hence,  CIA  =  ^  {C  +  B). 

Since  ^C  is  equal  to  AF,  the  angle  AFC  is  equal  to  the 
angle  C ;  hence,  the  angle  B  plus  FAB  is  equal  to  G; 
or  i^.1^  is  equal  to  C  —  B.  But  /6W  =  is  equal  to  one- 
half  of  FAB', 

hence,  ICH  =  \  {C  -  B), 


PLANE    TRIGONOMETKY. 


43 


Since  the  angle  ECI  is  inscribed   in  a  semicircle,   it  is  a 

right  angle    (B.   Ill,   P.   XVIII,    C.   2)  ;  hence,    CE    is    per- 

pendicular   to  CI,   at   the   point  C.      But  since   ///  is  parallel 

to   CE,   it  Avill   also   be   perpendicular  to  CI. 

I'roni   the  two    right-angled    triangles  ICE   and  ICH,   we 
have   (Formula  3,   Art.  37), 

EG  =  IG   tan  ^((7  +  7?),      and      IH  =  IG   tan  ^(C  -  i?); 

hence,   from   the   preceding   equations,  we  have,   after   omitting 
the    equal   factor    IG    (B.  11.,   P.  VII.), 


BG 


in 


tan  UG  -{-  E)      :     tan  i{G  -  B). 


The  triangles  ECB  and  lUB  being  similar,  their  hoino- 
logous  sides  are  proportional  ;  and  because  EB  is  equal  to 
AB  +  AG,  and  IB  to  AB  -AG,  we  shall  have  the 
proportion, 

EG     :    in    :  :    AB  +  AG     :    AB  -  AG. 

Combining  the  preceding  proportions,  and  substituting  for 
AB     and    AG     their   representatives    c    and     b,     we  have, 

c  +  b     :     c-b    ::     tanl(C+i?)     ;    tan^((7-J5)   .     .     (14.) 

Hence,  we    have    the   following    principle  : 

In  any  pla7ie  triangle,  the  sum  of  the  sides  iiicluding 
either  angle,  is  to  their  dlffereJice,  as  the  tajigent  of  half 
the  sum  of  the  tioo  other  angles,  is  to  the  tangent  of  half 
their   difference. 

Tlie   half  sum   of  the   angles  may  be   found  by  subtracting 
the  gi-^un  angle  from   180°,  and  dividing  the  remainder  by  2 
the   half   difference   may  be    found   by  means   of   the   principle 
just   demonstrated.        Knowing   the    half    sum     and    the    lialf 


M    '  PLANE      TRIGONOMETRY. 

difference  the  greater  angle  is  found  by  adding  the  half 
difference  to  the  half  sum,  and  the  less  angle  is  found  by 
subtracting  the  half  difference  from  the  half  sum.  Then  the 
solution  is   completed   as  in   Case   I. 

EXAMPLES. 

1.     Given     c  =  540,      h  —  450,      and     A  =  80°,     to  find 
Bf     Cy      and      a. 

OPERATION. 

c  +  b  -  990;     c  -  5  =  90  ;     ^  ((7+i?)  =  i(180°  -  80°)  =  50°. 

Applying   logarithms   to   Formula   (14),   vre   have, 

(a.  c.)  log  {c+  b)  +  log  (c  -  h)  +  log  tan  I  {0  +  B)  -  10  = 

log   tan  I   {C  —  D). 

(a.  c.)  log  [c  +  h)     .    .     (990)  7.004365 

log  {c  -h)     .     .     ( 90 )  1.954243 

logtaui(C+^)    (50°)  10.076187 

log   tan   \    {0  -  B)  9.034795  .-.  i  {C-B)  =  Q°  11'; 

C  =  50°  +  6°   11'  =  56°   11';     B  =  50°  -  6°   11'  =  43°   49'. 

From  Formula   ( 13 ),   we   have, 

sin   C  :   sin   ^   :   :  c  :  a;    whence, 

(ft.  c.)  log   sm  C    (56°  11')      .  0.080492 

log  sin  A        (80°)     .    .  9.993351 

log  c  .    .    .     (540)    .    .  2.732394 

log  rt 2.806237    .*.    a  =  640.082. 

Am.     B  =  43°  49',      C  =  56°  11',      a  =  640.082. 


PLANE      TRIGONOMETRY. 


45 


2.  Given    c  =  1686  yds.,     b  =  960  yds.,  and  A  —  128°  04', 
to   find     JB,     C,      and    a. 

Ans.     7?  =  18°  21'  21",     C  =  33°  34'  39",     a  =  2400  yds. 

3.  Given       a   =    IS.'ZSg  yds.,  b   =    7.642  yds.,        and 
a  -  45°  18    28",      to   find    A,     B,     and     c. 

Ans.    A  z=  112°  34'  13",     B  —  22°  OV  19",     c  =  14.426  yds 

4.  Given        a   =    464.7  yds,  b    =    289.3  yds.,         and 
6    =    87=  03'  48",      to  find     A,     B,     and     c. 

Alls.     A  =  60°  13'  39",     B  =  32°  42' 33",     c  =  534.66  yds. 

5.  Given        a    =    16.9584  ft.,         b    =    11.9613  ft.,        and 
C  =    60°  43'  36",      to   find     A,      B,      and      c. 

Ajis.     A   =   76°  04'  10",     B  =  43°  12'  14",     c  =  15.22  ft. 

6.  Given     a  =  3754,     b  =  3277.628,    and    C  =  57°  53'  17", 
to  find    A,    B,     and     c. 

Ans.    A  =  68°  02'  25",      B  =  54°  04'  18",     c  =   3428.512. 


CASE     rv. 


Given  the  three    sides   of  a    triangle.,   to  find   the  remaining 

parts.* 

46.  Let  ABC  represent  any 
plane  triangle,  of  which  BC  is 
the  longest  side.  Draw  AD  per- 
pendicular to  the  base,  dividing  it 
into  two  segments    CD     and    BJD. 


*  The   angles  may  be   found  by   Formula    {-^)    or    (ij)),    Lemma.      Pages 

109,   and  110,   Mensuration. 

20 


46 


PLANE    TRIGONOMETRY. 


From    the    riglit-angled    triangles     CAD    and    BAD,    W6 
have, 

and 


AD^  =  AC"  -  DC-, 


AD' 


AB'  -  BD  \ 


Equating  these  vahies  of  AD^,   we  have, 


AC  -  DC    =  AB'  -  BD  \ 
wlience,   hy  transposition, 


AC  -  AB'  =  DC  -  BD\ 

Factoring   each  member,   we  have, 

{AG  +  AB)  {AG  -  AB)    =    {DC  +  BD)  {DC  -  BD). 

Converting    this    equation    into    a    proportion    (B.  11.,   P.  11.), 
we  have, 
DG  +  BD    :    AG  +  AB    ::    AG  -  AB    :    DG-BD; 

or,    denoting    the    segments  by      s     and     s',      and    the    sides 
of  the   triangle  by     a,      5,     and      c. 


s  +  s' 


b  +  c 


b  -c 


s  —  s' 


(15.) 


that  is,  if  in  any  plane  triangle,  a  line  be  drawn  fi'om  the 
vertex  of  the  vertical  angle  perpendicular  to  the  base,  di^id- 
ing   it   into  two  segments  ;    then, 

The  sum  of  the  two  segments,  or  the  whole  base,  is  to 
the  sum,  of  the  two  other  sides,  as  the  difference  of  these 
sides  is   to   the  difference  of  the  segments. 

The  half  difference  added  to  the  half  sum,  gives  the 
greater,  and  the  half  difference  subtracted  from  the  half  s-jna 
gives  the  less  segment.  We  shall  then  have  two  riglit- 
fingled  triangles,  in  each  of  which  we  know  the  hypothenuse 
and  the  base  ;  hence,  the  angles  of  these  triangles  may  be 
found,  and   consequently,  those   of  the   given  triangle. 


PLANE      TRIGONOMETRY.  47 

EXAMPLES. 

1.     Given     a  =  40,     b  =  34,     and     c  ^   25,     to   find     A^ 
B,      and     C. 

OPERATION. 

Applying   logarithms   to   Formula    (15),   we   have, 

(a.  c.)  log  (s  +  s')  +  log  (J  +  c)  4-  log  {d  -  c)  =  log  (*  -  s') ; 

(a.  c.)  log  (s  +  s')     .  .    (40)     .    .  8.397940 

log  {b  +  c)     .  .     (59)     .     .  1.770852 

log  {b  -  c)     .  .     (9)     .     .  0.954243 

log  {s  -  s') 1.123035   .-.   s-s'  =  13.275. 

s   =  ^{s  +  s')  +  ^{s  -  s')   =  26.G375 

s'  =  i{s  +  s')  -  I-  (s  -  s')   =  13.3625 

From  Formula   ( 11 ),  we  find, 

log  s  +  (a.  c.)   log  5  =  log  cos  C    .  • .     C=  38°  25'  20",     and 
log  5'  +  (a.  c.)   log  c  =  log  cos  5    .  • .     B=  57°  41'  25" 

96°  06'  45" 


A    =    180'  -  96°  06'  45"  =i  83°  53'  15". 

2.  Given      a  =  6,       5  =   5,      and      0  =  4,      to   find     A, 
fi,      and     a 

Ans.     A  =  82°  49'  09",    J3  =  55°  46'  16",    C  =  41°  24'  35" 

3.  Given      a  =  71.2  yds.,     b  =  64.8  yds.,     and      c  =  31. 
yds.,    to   find    A,      J3,     and     C. 

Ans.     ^  =  83°  44' 32",    ^  =  64°  46' 56",     C  =  31°  28' 30" 


48 


PLANE     TRIGONOMETRY. 


PROBLEMS. 

1.  Kno'^'Uig  tlie  distance  A  J), 
equal  to  600  yards,  and  the  angles 
BA  0  ^  57°  35',  ABC  ^  64°  51', 
til  id  the  two  distances  AC  and 
BG. 

Ans.    AC  z=  643.49  yds.,      BC  =  600.11  yds. 

2.  At   what    horizontal   distance   from   a    column,    200    feet 

high,   will   it   subtend   an    angle    of    31°   17'  12"? 

Ans.     329.114  ft, 

3.  Required  the  height 
of  a  hill  D  above  a  hor- 
izontal plane  AJj,  the  dis- 
tance between  A  and  JJ 
being  equal  to  975  yards, 
and  the  angles  of  elevation  at  ^1  and  D  being  respect- 
ively   15°  36'     and     27°  29'.  Ans.     DC  =  587.61  yds. 


4.  The  distances  AC  and  DC 
are  found  by  measurement  to  be,  res- 
pectively, 588  feet  and  672  feet,  and 
their  included  angle  55°  40'.  Requir- 
ed  the   distance     A  JO. 

Ans.     592.907  ft. 


5.  Being  on  a  horizontal  plane,  and  wanting  to  ascertain 
the  height  of  a  tower,  standing  on  the  top  of  an  inaccessible 
hill,  there  were  measured,  the  angle  of  elevation  of  the  top 
of  the  hill  40",  and  of  the  top  of  the  tower  51°  ;  tlien 
raeasurmg   ui   a   direct   line    180   feet  farther  from  the  hill,   the 


PLANE     TRIGONOMETRY. 


49 


angle    of   elevation    of  the    top    of   the    tower    was    33°  45'  ; 
required  the  height   of  the  tower.  Ans.     83.998  ft. 


6.     Wanting  to   know  the  horizontal  distance  between   two 
inaccessible   objects  -C    and  W,   tlie 


following  measurements  were  made  : 


AH  =  536     yards 

J3AW  =  40°  16' 

WAB  z=z  57°  40' 

AHJS  =  42°  22' 


VIZ  :  < 


EBW 


Vl°  07'. 


Required  the   distance    JEW. 


Ans.     939.634  yd& 


F\-y 


7.  Wanting  to  know  the 
horizontal  distance  between 
two  inaccessible  objects  A 
and  _5,  and  not  finding  any 
station  fioni  which  both  of 
them  could  be  seen,  two 
points  C  and  Z>,  were  chosen 
at  a  distance  from  each  other 

equal  to  200  yards  ;  from  the  former  of  these  points,  A 
could  be  seen,  and  from  the  latter,  B  ;  .nnd  at  each  of  the 
points  C  and  Z),  a  staff  Avas  set  up.  From  C,  a  dis- 
tance CF  was  measured,  not  in  the  direction  Z>C,  equal 
to  200  yards,  and  from  2>,  a  distance  DE^  equal  to  200 
yards,   and   the   following   angles   taken  ; 


AFC  =     83°  00',         BBF  =   54°  30', 
BBC  =  156°  25',  ACF  =  54°  31', 

Required  the   distance    AB. 


ACB  =  53°  30' 

BED  =   88°  30' 

Ans.     345  467  yda. 


50 


PLANE      TRIGONOMETRY. 


8.  The  distances  J.i?,  AC,  and 
BC,  between  the  pomts  A,  Ji,  and 
(7,  are  kno^\^l  ;  viz.  :  AB  =  800  yds., 
AC  =  600  yds.,  and  J) C  =  AOO  yds. 
From  a  fourth  pomt  P,  the  angles 
AFC  and  BPC  are  measured  ; 
viz.  :  APC  =   33°  45', 

and  BFC  =  22°  30'. 

Required  the   distances    AP,     JBF,     and    CP 

{  AP 
Ans. 


p--. 


710.193  yds. 
BP   =     934.291  yds. 
^  CP  —  1042.522  yds. 


This  problem  is  used  in  locating  the  position  of  buoys  in 
maritime  surveying,  as  follows.  Three  points  A,  B,  and 
(7,  on  shore  are  known  in  position.  The  surveyor  stationed 
at  a  buoy  P,  measures  the  angles  APC  and  BPC.  The 
distances    AP,     BP,     and     CP,     are   then   found   as  follows  : 

Suppose  the  circumference  of  a  circle  to  be  described 
through  the  points  A,  B,  and  P.  Draw  CP,  cuttmg 
the  circumference   in    D,    and   draw  the   lines    DB     and    DA. 

The  angles  CPB  and  DAB,  being  inscribed  in  the 
same  segment,  are  equal  (B.  III.,  P.  XVIII.,  C.  1)  ;  for  a 
like  reason,  the  angles  CPA  and  DBA  are  equal  :  hence, 
in  the  triangle  ADB,  we  know  two  angles  and  one  side  ; 
we  may,  therefore,  find  the  side  DB.  In  the  triangle  ACB, 
we  know  the  three  sides,  and  we  may  compute  the  angle  B. 
Subtractuig  from  this  the  angle  DBA,  we  have  the  angle 
DEC.  Now,  in  the  triangle  DBC,  we  have  two  sides 
and  their  included  angle,  and  we  can  find  the  angle  DCB. 
Finally,  in  tlie  triangle  CPB,  we  have  two  angles  and  one 
side,  from  which  data  we  can  find  CP  and  BP.  In  like 
manner,   we   can    find    AP. 


ANiLYTICAL    TIUGONOMETRY. 


47.     Analytical  Trigonometey  is  tliat    branch   of   Mathe- 
matics  which    treats    of   the    general    properties    and   relations 


of  trigonometrical   functions. 


DEFIKITIONS     AND     GENERAL     PRINCIPLES. 


48.  Let  ABCD  represent  a  cir- 
cle whose  radius  is  1,  and  suppose 
its  circumference  to  be  divided  into 
four  equal  parts,  by  the  diameters 
AC  and  BD^  drawn  perpendicular  to 
each  other.  The  horizontal  diameter 
A  C,  is  called  the  initial  diameter  ; 
the   vertical    diameter    BB^     is    called 

the  secondary  diameter  ;  the  point  J.,  from  which  arcs  are 
usually  reckoned,  is  called  the  origin  of  arcs,  and  the  point 
B,  90°  distant,  is  called  the  secondary  origin.  Arcs  esti- 
mated from  A,  around  towards  B,  that  is,  in  a  direction 
contrary  to  that  of  the  motion  of  the  hands  of  a  watch,  are 
considered  positive ;  consequently,  those  reckoned  in  a  con 
trary  direction   must   be   regarded   as    negative. 

The  arc  AB,  is  called  l\ie  first  quadrant;  the  arc  BCy 
the  second  quadrant ;  the  arc  CD,  the  tliird  quadrant ; 
and  the  arc    DA,    the  fourth  quadrant.       The  point  at  which 


52 


ANALYTICAL 


an   arc    terminates,  is   called  its   extremity^  and  an  arc  is  said 
to   be   in    that    quadrant    in   which    its    extremity   is    situated. 
Thus,    the   ai'c    AM     is    in    the  first 
quadrant,  the   arc    A3f'    in  the   sec- 
ond^   the    arc    A3I"      in    the    third, 
and  the   arc    AM'"    in   the  fourth. 


49.  The  comjjlement  of  an  arc 
has  been  defined  to  be  the  dificr- 
ence  between  that  arc  and  90°  (Art. 
23)  ;     geometrically     considered,     the 

complement  of  an  arc  is  the  arc  included  hetween  the  extremity 
of  the  arc  and  the  secondary  origin.  Thus,  MB  is  the 
complement  of  A3I  ;  31' B,  the  complement  of  A3I' ; 
31" B,  the  complement  of  A3I",  and  so  on.  When  the 
arc  is  greater  than  a  quadrant,  the  complement  is  negative, 
according  to  the  conventional  pi'inciple   agreed  upon   (Art.  48). 

The  suppleynent  of  an  arc  has  been  defined  to  be  the 
difierence  between  that  arc  and  180°  (Art.  24)  ;  geometrically 
i;(jnsidered,  it  is  tJte  arc  included  between  the  extremity  of 
the  arc  and  the  left  hand  extremity  of  the  initial  diameter. 
Thus,  3IC  is  the  supplement  of  A3I,  and  M"C  the  sup- 
plement of  A3I''.  The  supplement  is  negative,  when  the 
arc   is   greater   than   two   quadrants. 


■  50.  The  sine  of  an  arc  is 
the  distance  from  the  initial 
diameter  to  the  extremity  of  the 
arc.  Thus,  PM  is  the  sine 
of  A3r,  and  P"3I"  is  the 
sine  of  the  arc  A3I".  The 
term  distance,  is  used  in  the 
sense  of  shortest  or  perpendicu- 
lar distance. 


TRIGONOMETRY.  63 

51.  The  cosine  of  an  arc  is  the  distance  from  the  sec- 
ondary diameter  to  the  extremity  of  the  arc  :  thus,  JVM 
ia   tliG   cosine   of   AM,    and    JVM'     is   the   cosine   of   AM'. 

The  cosine  may  be  measured  on  the  initial  diameter  : 
thus,  OP  is  equal  to  the  cosme  of  A^l,  and  OF'  to  tlio 
cosine   of   AM'. 

52.  2^ he  versed-sine  of  ayi  arc  is  the  distance  from  the 
sine  to  the  origin  of  arcs  :  tluis,  PA  is  the  versed-sine  of 
AM,     and    P'A    is  the   versed-siue   of   A3I'. 

53.  The  co-versed-sine  of  an  arc  is  the  distance  from 
the  cosine  to  the  secondary  origin  :  thus,  JVP  is  the  co- 
versed-sme  of  AM,     and  2f"B  is  the  co-versed-sine  of  AM". 

5i.  The  tangent  of  an  arc  is  that  part  of  a  perpen- 
dicular to  the  initial  diameter,  at  the  origin  of  arcs,  in- 
cluded between  the  origin  and  the  prolongation  of  the  diam- 
eter through  the  extremity  of  the  arc  :  thus,  AT  is  the 
tangent  of  AM,  or  of  AM",  and  AT"  is  the  tangent 
of    AM',      or   of    AM'". 

55.  The  cotangent  of  an  are  is  that  imrt  of  a  perpcrtr 
dicxdar  to  the  secondary  diameter,  at  the  secondary  origin, 
included  hetxoecn  the  secondary  origin  and  the  prolongation 
of  the  diameter  through  the  extremity  of  the  arc  :  thuR, 
BT'  is  the  cotangent  of  AM,  or  of  AM",  and  BT  ' 
is  the   cotangent   of   AM',     or   of   AM'". 

56.  The  secant  of  an  arc  is  the  distance  from  the  ce?> 
tre  of  the  arc  to  the  extremity  of  the  tangent :  thus,  OT 
is  the  secant  of  A3T,  or  of  A3I",  and  OT'"  is  the  se- 
cant   of   AM',     or   of    AM'". 

67.     The  cosecajit    of   an    arc    is    the    distance  from    the 


64 


ANALYTICAL 


centre  of  the  arc  to  the  extremity  of  the  cotangent  :  thus, 
OT'  is  the  cosecant  of  A3I,  or  of  AM'\  and  OT"  is 
tlie  cosecaut   of   A3I',     or   of   AM'". 

The  term,  co,  iu  combination,  is  equivalent  to  complement 
of  j  tlius,  the  cosine  of  an  arc  is  the  same  as  the  sine  of 
the  comjylement  of  that  arc,  the  cotangent  is  the  same  as 
the   tangent   of  the  complement^   and  so   on. 

The  eight  trigonometrical  functions  above  defined  are  also 
called  circular  functions. 

KULES     FOK     DETERMINING     THE     AiGEBEAIC     SIGNS     OF    CLRCULAK 

FUNCTIONS. 


58.  All  distances  estimated  upwards  are  regarded  as  p>os- 
itive  /  consequently^  all  distances  esti7nated  downwards  must 
he  considered  negative. 

Thus,    AT,   PM,    NB,   P'M\  r^ 

are  positive,   and  AT"\    P"'M"\    ^' 
P"M'\     &c.,    are   negative. 

All  distances  estimated  towards 
the  right  are  regarded  as  positive  / 
consequently,  all  distances  estimat- 
ed toicards  the  left  must  be  con- 
sidered negative. 

Thus,  iO/,  HT',  PA,  &c., 
are   positive,   and    JST'M',     BT",     &c.,   are  negative. 

All  distances  estitnated  from  the  centre  in  a  directio?i  to 
towards  the  extremity  of  the  arc  are  regarded  as  j^ositive  ; 
consequently,  all  distances  estimated  in  a  direction  from  the 
second  extremity   of  the  arc  must  he  considered  negative. 

Thus,  OT,  regarded  as  the  secant  of  x\3[,  is  estimated 
in   a   direction   towards    M,      and   is  positive  ;    but     OT,      re- 


TRIGONOMETRY.  55 

garded  as   the  secant   of   AM'\      is   estimated    iu   a   direction 
from    J/",     and   is  negative. 

These  conventional  rules,  enable  us  at  once  to  give  the 
proper   sign   to   any   function   of  an   arc   in   any   quadrant. 

59.  In  accordance  with  the  above  rules,  and  the  defini- 
ions  of  the  circular  functions,  we  have  the  following  princi 
)Ies  : 

The  sine  is  positive  in  the  first  and  second  quadrants^ 
and  negative  in   the  third  and  fonrth. 

The  cosine  is  positive  in  the  first  and  fourth  quadratits^ 
and  negative  in   the  second  and  third. 

The  versed-sine   and  the  co-versed-sine  are   alioays  positive. 

The  tangent  and  cotangent  are  positive  in  the  fir^t  and 
third  quadrants^   and  tiegative  in  the  second  and  fourth. 

The  secant  is  p>ositive  in  the  first  and  fourth  quadrants^ 
a  I'd  negative  in   the  second  and  third. 

The  cosecant  is  positive  in  the  first  and  second  quadrants, 
and  negative   in  the  third  and  fourth. 


LIMITING     VALUES     OF     THE      CIRCULAR     FUNCTIONS. 

60.  The  limitmg  values  of  tlie  circular  functions  are  those 
values  which  they  have  at  the  beginnuig  and  end  of  the 
different  quadrants.  Their  numerical  values  are  discovered 
by  following  them  as  tlie  arc  increases  from  0°  around  to 
3G0°,  and  so  on  around  through  450°,  540°,  &c.  The  signs 
of  these  values  are  determined  by  the  principle,  that  the  sign 
of  a  varying  magnitude  up  to  the  limit,  is  the  sign  at  the 
limit.  For  illustration,  let  us  examine  the  limiting  values  of 
the   sine   and   tangent. 


56  ANALYTICAL 

If  Ave  suppose  the  arc  to  be  0,  tlie  sine  will  be  0  ;  as 
tlie  arc  increases,  the  sine  increases  until  the  arc  becomes 
equal  to  90°,  when  the  sine  becomes  equal  to  +1,  -which  is 
its  greatest  possible  value ;  as  the  arc  increases  fiom  90°, 
the  sine  goes  on  diminishing  until  the  arc  becomes  equal  to 
180°,  when  the  sine  becomes  equal  to  +  0  ;  as  the  arc 
increases  from  180°,  the  sine  becomes  negative,  and  goes  on 
increasing  numerically,  but  decreasing  algebraically^  until  the 
arc  becomes  equal  to  270°,  when  the  sme  becomes  equal  to 
—  1,  which  is  its  least  algebraical  value  ;  as  the  arc  increases 
from  270°,  the  sine  goes  on  decreasing  numerically,  but  in- 
creasing algebraically^  until  the  arc  becomes  3C0°,  when  the 
sine  becomes  equal  to  —  0.  It  is  —  0,  for  this  value  of 
the   arc,   in   accordance   with   the   jirinciple   of  limits. 

The  tanerent  is  0  when  the  arc  is  0,  and  increases  till 
the  arc  becomes  90°,  when  the  tangent  is  +  co  ;  in  passing 
through  90°,  the  tangent  changes  from  +  co  to  —  a;  ,  and 
as  the  arc  increases  the  tangent  decreases,  numerically,  but 
increases  algebraically,  till  the  arc  becomes  equal  to  180°, 
when  the  tangent  becomes  equal  to  —  0  ;  from  180°  to 
270°,  the  tangent  is  again  positive,  and  at  270°  it  becomes 
equal  to  +  co  ;  from  270°  to  360°,  the  tangent  is  agam 
negative,    and   at    3C0°   it   becomes   equal   to     —  0. 

If  we  still  suppose  the  arc  to  increase  after  reaching  360°, 
the  functions  will  again  go  through  the  same  changes,  that 
is,  the  functions  of  an  arc  are  the  same  as  the  functions 
that    are   increased   by  360°,    720°     &c. 


By   discussing   the  limiting   values   of   all   the  circular   func 
tions  we   are   enabled   to   form   the   following   table  : 


TRIGONOMETRY. 


67 


T  A  BLE       I, 


Arc  =•  0. 

Arc  = 

90°. 

Arc  = 

180°. 

Arc  = 

27u°. 

Arc  =  SCO°. 

sin          ~  0 

sin 

=   ] 

siu 

=     0 

sin 

=  -1 

ain         =—0 

C08           =    1 

cos 

=   0 

cos 

«=-! 

cos 

=  -0 

cos        =     1 

v-sin       =  0 

v-sin 

=    1 

v-sin 

=     2 

v-sin 

=      1 

v-sin     =     0 

co-v-sin  =   1 

co-v-sin 

=    0 

co-v-sin 

=      1 

co-v-sin 

=     2 

c-v-sin  =     1 

tan          —  0 

tan 

=  oc 

t:Ul 

=  -0 

tan 

=       CO 

tan        =  —0 

cot              =   CO 

cot 

=   0 

cot 

=  — CO 

cot 

=     0 

cot           =  —  CO 

sec         —   1 

sec 

=  cc 

sec 

=  -1 

sec 

=  — CO 

sec        =     1 

cosec      =  CO 

cosec 

=  1 

cosec 

=      CO 

cosec 

=  -1 

cosec    =  — CO 

RELATIONS     BETWEEN     THE      CIRCULAR     FUNCTIONS      OF      ANY     ARC. 


61.  Let  A3f  represent  any  arc  de- 
noted l)y  a.  Draw  the  lines  as  repre- 
sented in  tlie  fiorure.  Tlien  we  sliall 
have,   hy   definition 


OM  =  OA  =  1  ;      PJ/  =  OJSr  = 
XA[  ^  OF  ^  cos  a 

NB      : 

BT'  =  cot  a 


sni  a 


co-ver-sin    a 


PA    =  ver-siu  a  ; 
AT  =    tan    a ; 


OT  =.  sec  « 


and     OT'  z=  cosec  a. 


From  tlie   right-angled   triangle    OPM^     we   have, 


PM^  -f  OP""  =  OJ/' , 


or. 


sin^a  -f-  cos^a  =  1.       .       (1.^ 


Tiie   symbols    sin^a,      cosher,      &c.,    denote   the  square    of   the 
sine   of    a,"   the   square   of  the    cosine   of    a,     &g. 
From   Formula   ( 1 )    we   have,  by   transposition, 


sin'a 


1  —  cos^a     .     (2)  ;      and     cos-a  =  1  —  sin^a.     .     (3.) 


58  ANALYTICAL 

We  have,   from   the   figure, 
FA  =  OA-  OF, 
or,  ver-sin  a  =  I  —  cos  a.     ,     .     (4.) 

and,  NB  =  OB  -  OK, 

or,       co-ver-sm  a  =  1  —  sin  a.      .     .  .  (5.) 


'1 


From   the   similar  triangles     OAT     and     0PM,     we   have, 

OF  :  FM  :  :  OA  :  AT,     or,     cos  a  :   sin  a    :  :    1    :    tan  a ; 

sin  a  ,  ^  . 

whence,  tan  a  =  (6.) 

'  cos  a 

From  the   similar   triangles    OJSTJI    and    OBT',    we  have, 

ON  :  NM :  :  OB  :  BT',     or,     sin  a   :   cos  a  :  :    1   :   cot  a ; 

cos  a  ,  ^  . 

whence,  cot  a  =  — (  /.) 

'  sm  a 

Multiplying   ( 6 )    and   ( 7 ),    member  by  member,   we  have, 

tan  a   cot  a  =  1  ;       (8.) 

whence,  by  division, 

tan  a  =  ;      •     ( 9.)       and       cot  a  =  •     •     ( 10.) 

cot  a  ^  tan  a 

From   the   similar   triangles     0PM    and    OAT,     we  have, 
OP   :    OM   :  :   OA    :   OT,      or,     cos  a    :    1    :  :    1    :    sec  a 

whence,  sec  a  =    (H) 

cos  a 


TRIGONOMETRY.  59 

From   the   similar   triangles     ONM    and    OBT\     we  have, 

ON  :   OM  \  '.   OB    \   0T\    or,    sin  a   :    1    :  :    1    :  co-sec  a; 

whence,  co-sec  a  —  —. —  •         •     •         •  12.) 

sm  a  ' 

From   the   right-angled   triangle    OAT^     "we   have, 


OT'  -   OA'  4-  AT;      or,       sec^a  =  1  -f  tan^a.      .    (13.) 

From   the   right-angled  triangle    OBT\     we   have, 
07^'  =   OB'  +  BT'';     or,    co-sec^a  =  1  +  cot^a.     .  (14.) 

It  is   to   be   observed  that  Formulas  (5),    (7),    (12),    and 

(14),    may  be   deduced   from  Formulas  (4),    (6),    (11),   and 

( 13 ),  by  substituting   90°  —  a,     for    a,  and  then  making  the 
proper   reductions. 

Collecting  the  preceding  Formulas,  we  have  the  following 
table  : 

TABLE       II. 


(1.) 

(2.) 

sin'a  +  cos'a 
sin'a 

= 

1. 

1  —  cos'a. 

(9.) 

tan  a 

= 

1 

cot  a 

(3.) 

cos'a 

= 

1  —  sin'a. 

(10.) 

cot  a 

- 

1 

tan  a  ' 

(4.) 

ver-sin  a 

= 

1  —  COS  a. 

1 

(5.) 

co-ver-sin  a 

= 

1  —  sin  a. 

(11.) 

sec  a 

= 

cos  a 

(6.) 

tan  a 

= 

sin   a 

(12.) 

cosec  a 

^ 

1 
sin  a 

cos  a 

(V.) 

cot  a 

= 

cos  a 

(13.) 

sec'a 

= 

1  -ftan'a. 

sin   a 

j     (8) 

I 

tan  a   cot  a 

= 

1. 

(14. 

coscc'a 

= 

1  4-  cot'a. 

CO 


ANALYTICAL 


FUNCTIONS    OF    NEGATIVE    ARCS. 


62.     Let     A3I"\       estimated    from    A    towards    D^      be 
numerically  equal  to    A3f ;    then, 
if  Ave  denote  the  arc  AM    by    a,     -p/' 
the  arc     AM'"     will    be   denoted 
by    —  a     (Art.    48). 

All    the    functions    of    A3I"\ 
will    be     the     same     as    those     of 
AB3I'"  \    that  is,  the  functions  of 
—  a     are    the    same    as    the    func- 
tions  of    360°  —  a. 

From   an  inspection   of  the   fig- 
ure, we   shall   discover    the  following   relations,   viz.  : 

sin  (—  a)  =  —  sin  a  ;  cos  (—  a)  =  cos  a  ; 
tan  (—a)  =  —  tan  a  ;  cot  (—  a)  =  —  cot  a  ; 
see  (—  a)    =        sec  a  ;         cosec(—  a)     =    —  cosec  a. 


rn/t 

] 

B 

T 

N.   A/T'/^'^ 

N 

X^JT' 

P'        \ 

\ 

A 

0 

V 

1 

V 

'>. 

i 

'-tmh 

FUNCTIONS    OF    AKCS    FORMED    BY    ADDING    AN    ARC    TO,    OR    SUB- 
TRACTING   IT    FROM    ANY    NUMBER    OF    QUADRANTS. 

63.     Let     a    denote   any   arc  less    than    90°.      From  what 
has  preceded,   we   know   that, 

sin  (90°  —  a)    =    cos  a  ;  cos  (90°  —  a)       =    sin  a. 

tan  (90°  —  a)    =    cot  a  ;  cot  (90°  —  a)       =    tan  a. 

B8C  (90°  —  a)    z=    cosec  a  ;         cosec  (90°  —  a)     =    sec  a. 

Now,  suppose  that  JjM'  =  a,    then  will  AM'  =  90°  +  a. 
We  see  from  the  figure  that, 


TRIGONOMETRY.  61 

KM'   =   sin  a,        P'Jf'  =    cos  a,        J3T"  =   tan  a, 
^r'"  =   cot  «,        or"  =   sec  a,         07""  =   coseo  a^ 
without  reference  to  their  eigns. 

By   a    simple  inspection    of   the   figure,   observing   the   rul 
for  signs,  we  deduce  the  following  relations : 

siu  (90°  +  a)    =       cos  a,  cos  (90°  +  a)  =  —  sin  a, 

tan  (90°  +  a)    =  —  cotan  a,         cot  (90°  +  a)  =  —  tan  a, 

sec  (90°  +  «)    =  —  coseo  a,         cosec  (90°  +  a)       =       sec  a. 

Again,   suppose 

31' C  =  AM  =  a  ;      then  will    A3I'  =  180°  -  a. 

We   see  from  the   figure  that, 

P'M'  =   sin  «,         OP'  =   cos  a,  ^^"'  =   tan  a, 

i?r"   =   cot  a,        OT"  =   sec  «,  0^"'  =   cosec  a, 

without  reference  to  their  signs  :   hence,   we  have,  as  before, 
the  following   relations  : 


o 


sin  (180°  ~  a)    =       sin  a,  cos  (180°  —  a)     —   —  cos  a, 

tan  (180°  —  a)    =  —  tan  a,  cot  (180°  —  a)     =   —  cot  a, 

sec  (180°  —  a)    =  —  sec  a,  cosec  (180  —  a)   —       cosec  a, 

By   a  smiilar    process,   we  may   discuss  the   remaining   arcs 

in   question.       CoUectmg    the  results,   we    have    the   following 

table  : 

21 


62 


ANALYTICAL 


TABLE       III. 


Bin   = 


itan  =  — 


Arc  =  90°  +  a. 

cos  a,       cos      =  —  sin  a, 
cot  a,       cot      =  —  tan  a, 


Isec 

=  —  cosec  a, 

cosec  = 

sec  a. 

Arc   — 

180°  -  a. 

sin 

—       sin  a, 

cos      =  — 

cos  a, 

tan 

=  —  tan  a, 

cot      =  — 

cot  a, 

sec 

=  —  sec  a, 

cosec  = 

cosec  a. 

Arc  = 

180°  +  a. 

sin 

=  —  sin  a, 

cos      =  — 

cos  a, 

tan 

—       tan  «, 

cot      = 

cot  a, 

sec 

=  —  sec  a, 

cosec  =  — 

cosec  a. 

sm 
tan 
sec 


sin 
tan 
sec 


sin 
tan 
sec 


Arc  =  270°  -  a. 

cos  a,  cos      =  —  sin  a, 

cot  a,  cot      =       tan  a, 

cosec  a,  cosec  =  —  sec  a. 


Arc 


27  O'  4  a. 


cos  a,  cos      =       sin  a, 

cot  a,  cot      =  —  tan  a, 

cosec  a,  cosec  =  —  sec  a. 

Arc  =  360°  -  a. 

■  sin  a,  cos      =       cos  a, 

tan  a,  cot      =  —  cot  a, 

sec  a,  cosec  =  —  cosec  a. 


It  will  be  observed  that,  when  the  arc  is  added  to,  or 
subtracted  from,  an  even  number  of  quadrants,  the  name  of 
the  iimction  is  the  same  in  both  columns ;  and  when  the 
arc  is  added  to,  or  subtracted  from,  an  odd  number  of  quad- 
rants, the  names  of  the  ftmctions  in  the  two  columns  are 
contrary :  in  all  cases,  the  algebraic  sign  is  determined  by 
the   rules   already  given   (Art.  58). 

By  means  of  this  table,  we  may  find  the  functions  of 
any   arc  in  terms   of   the  functions   of   an    arc  less  than    90" 

Thus, 

sin  115°  =    sin  (  90°  +  25°)    =         cos  25°, 

sin  284°   =    sin  (270°  +  14°)    =    —  cos  14% 

sm  400°   =    sin  (360°  +  40°)    =         sin  40°, 

tan  210°   =    tan  (180°  +  30°)    =         tan  30° 


TRIGONOMETRY, 


63 


PARTICULAR    VALUES     OF     CERTAIN     FUNCTIONS. 

64.  Let  3fAM'  be  any  arc,  denoted 
by  2a,  M'M  its  chord,  and  OA  a 
radius  drawn  perpendicular  to  M'M:  then 
will  PM  =  PM\  and  AM  =  AM' 
(B.  m.,  P.  VI.).  But  PM  is  the  sine 
of   AM^      or,     PM  =  sin  a :    hence. 

sin  a    =    ^M'M ; 

that   is,   the  si7ie  of  an  arc  is  equal   to   one  half   the   chord 
of  twice   the   arc. 

Let    31' AM  =  60°  ;     then  will    AM  =  30°,     and    M'M 
will    equal   the   radius,    or     1  :    hence,   we   have, 

sin  30°  =    i  ; 

that  is,  the   sine   of  30°   is  equal  to   half  ^^^^  radius. 
Also, 

cos  30°   =    -v/l  —  sin-  30°   =  ^-y/F; 
hence, 

sin  30°     _       /f 
cos  30°     ~  V  3  * 


tan  30°   = 


Again,   let     M'AM  =    90°  :      then   will      AM  = 
3I'M  =  yr    (B.  v.,   p.  m.)  :    hence,   we   have, 


45",      and 


Also, 
hence, 


sin  45°   =  -^-y/^  ; 

cos  45°  =    -/l  -  sin'''  45°   =  ^  yT; 


sin  45° 
tan  45°  =   -—    =    1. 

cos  45° 


Many   other   numerical   values  might  be   deduced. 


u 


ANALYTICAL 


i 

y 

/ 

\. 

L 

\ 

df 

/y^r^ 

^ 

\ 

c 

L-  P'  A 

FORMULAS     EXPEESSIXG     KELATIOXS     BETWEEN"     THE     CIECULAK 
FUNCTIONS     OF     DIFFEREI^T     AECS. 

G5.  Let  MB  and  BA  represent  two  arcs,  liaTing  the 
common  radius  1 ;  denote  the  first  by 
a,  and  the  second  by  I) :  then,  3IA=a  +  b. 
From  3f  draw  MF  perpendicular  to  CA, 
and  MJV  perpendicular  to  CB ;  from 
AT  draw  JVF'  perpendicular  to  CA,  and 
ATL  parallel  to  AC. 

Then,  by  definition,  we   shall  have, 

F2I  =  sin  {a  +  h),    NM  =  sin  a,    and  02^  =  cos  a. 

From  the  figure,   we  have, 

F3f=  3IL  +  LP (1). 

Since  the  triangle  3ILX  is  similar  to  CF'N  (B.  IV^ 
P.  21),  the  angle  L3[jSf  is  equal  to  the  angle  F'CJV;  hence, 
from   the   right-angled   triangle   31  LN,  we   have, 

3IL  ■=  31N  cos   5  =  sin   a  cos  h. 

From  the  right-angled  triangle   CF'N  (Art.  37),  we  have, 

AT'  =  CN  sin   5 ; 

or,   since         NF'  =  LF,        LF  =  cos  a  sin   i. 

Substituting  the  values  of  F3f,  3IL,  and  LF,  in  Equar 
>ion    ( 1 ),   we  have, 

sin  (a  +  J)  =  sin  a  cos  5  -t-  cos  a   sin   h;    .     (^.)' 

that  is,  (he  sine  of  the  sum  of  two  arcs,  is  equal  to  the 
sine  of  the  first  into  the  cosine  of  the  second,  plus  the  co- 
sine of  the  first  into   the  sitie  of  the  second. 


TRIGONOMETRY.  66 

Since  tlie   above  formula  is  true  for  any  values   of    a    and 
5,   we   may   substitute     —  Z>,     for     h  ;     whence, 

sin  {a  —  h)    =    sin  a   cos  (  —  J)  +  cos  a  sin  (  —  6)  ; 
but    (Art.  G2), 

cos  {  —  b)  =  cos  5,         and,        sin  (  —  J)   =   —  sin  b  ; 

hence, 

sm  (a  —  b)   =  sin  a   cos  b  —  cos  a   sin  5  ;    •     ( 3.) 

that  is,  the  sme  of  the  difference  of  tioo  arcs,  is  equal  (c 
the  sine  of  the  first  into  the  cosine  of  the  second,  minus  the 
cosine  of  the  first  into  the  sine  of  the  second. 

If,  in  Formula  ( 3 ),  we  substitute    (90°  —  a),    for    a,    we 
have, 

Bin  (90°— a-i)  =  sin  (90°-a)  cos  5— cos  (90°— a)  sin  6 ;  •  (2.) 
but   (Art.  63), 

sin  (90°-  a  —  h)=  sin  [90°-  (a  +  5)]   =  cos  (a  +  b), 
and, 

sin  (90°  —  a)  =  cos  a,  cos  (90°  —  a)  =  sin  a  ; 

hence,   by   substitution   in   Equation    ( 2 ),   we   have, 

cos  {a  +  b)  —  cos  a   cos  6  —  sin  a  sin  &  ;     •     ( ®.) 

that  is,  the  cosine  of  the  sum  of  two  arcs,  is  equal  to  the 
rectanrjle   of  their  cosines,   minus  the  rectangle  of  their  sines, 

If;   in  Formula   (®),   we   substitute     —  b,    for    b,     we  find 

cos  {a  —  b)  =  cos  a  cos  (  —  5)  —  sin  a   sin  (  —  J), 
or, 

cos  {a  —  b)  =  cos  a   cos  &  +  sin  a   sin  J  ;     •     •     ( ll>.) 


66  ANALYTICAL 

that  is,  the  cosine  of  the  difference  of  two  arcs^  is  equcU 
to  the  rectanf/le  of  their  cosines^  ^;?<<s  tJie  rectanfjle  of  their 
sines.  • 

It  we  divide  Formula   ( <^ )    by  Formula   ( © ),  member  Ly 
i:eraber,   we   have, 

sin  [a  +  h)    _   sin  a  cos  h  4-  cos  a  sin  b 
cos  {a  +  b)         cos  a  cos  6  —  sin  a  sin  b 

Dividing  both  terms  of  the  second  member  by  cos  a  cos  ft, 
recollecting  that  the  sine  divided  by  the  cosine  is  equal  to 
the   tangent,  we   find, 

/      ,    7x  tan  a  +  tan  b  ,._.  . 

tan  {a  +  b)    = =^ r  ;    •    •    •    •    (a.) 

^      '      '  1  —  tan  a  tan  6  '  ^      ' 

that  is,  the  tangent  of  the  sum  of  two  arcs,  is  equal  to  the 
sum  of  their  tangents,  divided  by  1  7?ii?ius  the  rectangle  of 
their   tangents 


If,  in  Formula  ( 12 ),  we  substitute    —  5,     for    b,     recollect- 
ing that      tan  {—  b)  =  —  tan  5,      we  have, 

,         - .             tan  a  —  tan  b  ,  „ . 

tan  (a  —  b)    = j  ;••••(  ff.) 

^  ''  1  +  tan  a  tan  b  ^      ' 

that  is,  the  tangent  of  the  difference  of  two  arcs,  is  equal 
to  the  difference  of  their  tangents,  divided  by  1  ^lus  the 
rectangle  of  their   tangents. 


In   Uke  manner,   di\ading   Formula   ( ® )    by  Formula  ( ^ ), 
member  by   member,   and   reducing,   we   have, 

.    ,      ,    ,,            cot  a  cot  b  —  \  f^. 

cot  (a  +  b)    = ~  ;    •  •          (<a.) 

^           ''             cot  a  +  cot  6    *  ^      ' 


TRIGONOMETRY.  67 

and    theuce,   by  the   substitution   of    —  5,     for    J, 

^   ,         i>  cot  a   cot  &  +  1  ,      ^ 

cot  {a  —h)    = z ^^—  ;    .     .     .    .     (i2.) 

^  ''  cot  6  —  cot  a    '  ^     ' 

FDNCTIOKS  OF  DOUBLE  ARCS  AND  HALF  ARCS. 

66.     If,   in   Formulas    (^),     (©),     (a),     and     (Q),     we 
make    a  =  b,     we   find, 

sin   2a    z=    2  sin  «   cos  «;••••(  ^'.) 

cos  2«    =    cos^a  —  sin^a  ;••••(  ®'.) 

2  tan  a  ,     .  v 

tan  2a    =    — —  ; ( Q'.) 

1  -  tan^a    '  ^       ' 

^  „            cot^a  —  1  ,     . . 

cot  2a    = ( <3M 

2  cot  a  ^  ^  •' 

Substituting  in  ( ©'),   for    cos^a,     its  value,     1  —  sin^a ;      and 
afterwards   for     sin^a,     its  value,     1  —  cos^a,     we   have, 

cos  2a    =    1—2  sin^a, 
cos  2a    =    2  cos^a  —  1  ; 


whence,   by   soMng  these   equations, 


sm   a 


a     =    ^- 


'  1  —  COS  2a 

2            »     • 

»         •         • 

'  1  +  cos  2a 

•         •         • 

(1.) 


COS   a     =1^   ^ (2.) 

We   also   have,   from  the   same   equations, 

1  —  cos  2a    =    2  sin'a ; (3.) 

1  +  cos  2a    =    2  cos'a.      •         (4.) 


68  ANALYTICAL 

Dividing  Equation  ( ^'),    first  by  Equation  ( 4 ),   and   then 
by  Equation   ( 3  ),   member  by  member,   we  have, 

sin  2a              ^  .  ^  . 

=   tan  a ; o.) 


1  +  cos  2a 

sin  2a 
1  —  cos  2a 


cot  a,       (6.) 


/  1  —  cos  a 
V           2 

• 

1 

• 

» 

• 

• 

• 
• 

•         • 

/  1  +  cos  a 
V            2 

sin  a 

• 

•         « 

1  +  cos  a  * 
sin  a 

Substituting     |a,     for    a,      in  Equations   (  1 ),    (  2  ),    (  5  ), 
and    ( 6  ),     we  have, 

/I  —  CUB    u  f   r\  rr  \ 


cos   ia     =    s^  ■ ^ J      .     .     .     ( ©".) 

1                    sin  a  , .  ,,, . 

tan  ^a     =     -— ;       •    •    •    •     ('Ji" ) 


cot  la     =     , ( CsJ".) 

1  —  cos  a 


Taking   the   reciprocals   of   both   members    of  the   last   two 
formulas,   we   have   also. 


^   ,  1  +  cos  a  ,  ,  1  —  cos  a 

cot  *a  =■  7— ,         and,         tan  Aa  =   — ; 

sm  a  "  sm  a 


ADDITIONAL     FORMULAS. 

67.  If  Formulas  (^)  and  (3)  be  first  added,  member 
to  member,  and  then  subtracted,  and  the  same  operations  be 
performed   upon    ( © )     and    (  Si) ),    we   shall   obtain. 


TRIGONOMETRY. 


69 


sin  (a  +  5)  4-  sin  (a  —  b) 

sin  («  +  J)  —  sin  (a  —  b) 

cos  (a  +  J)  +  cos  (a  —  b) 

cos  (a  —  J)  —  cos  (a  +  5) 


2  sin  a  cos  b  ; 

2  cos  a  Bin  b  f 

2  cos  a  cos  b  ; 

2  sin  a  sin  b. 


If  in   these   we   make, 


a  -{-  b  =  Pi         and         a  —  5  =  g', 


whence, 


a  =  i{2^-\-  q), 


b  =  i  {p  -  q)  ; 


and  then   svibstitute  in  the   above   formulas,   we   obtain, 

sin  p  -{■  sin  q    =  2  sin  ^  {p  +  q)  cos  ^  {p  —  q)     •     ( ili.) 

sin  ^  —  sin  2'    =  2  cos  ^  (^  +  q)  sin  \  {p  —  q)     •     (  2».) 

cos 2^  +  cos  q    =  2  cos  i  {p  +  q)  cos  i  (i?  —  2')     •     (22.) 

cos  q  —  cosp    =  2  sin  ^  (/>  +  g-)  sm  I  (^  —  2')     •     (  SI.) 


From  Formulas   ( 2» )    and    ( JI2 ),    by  division,   we   obtain, 


sin  jt?  — sing-  _  cos  ^{p+q)  sin  ^{p—q)    __  ta.n^p—q) 
trnp  +  siaq  ~  sin^(^;-f2')  cos^{p—q)  ~  tan^(jt?  +  2') 


(1-) 


That  is,  the  sum  of  the  sines  of  two  arcs  is  to  their  dif- 
ference., as  the  tangent  of  one  half  the  sum  of  the  arcs  is 
to   the  tangent  of  one  half  their  difference. 


70  ANALYTICAL 

Also,   in   like  manner,   we   obtain, 

smp  +  sin  q  ^  sin  Hp+g)  cos|(p-g)   ^  ^.^^(^  ^    .    (2.) 
cos^  +  cosg-        con i{p+q)  COS i{p—q) 

mnp-sinq  ^  cos  Up  +  Q)  sin  Up-Q)  ^  tan  j  (^-g)     .     (3.) 
COEp  +  cosq       cos  i^{p+q)  cos  l{p  —  q)  ^  \jr     1/ 

sin^  +  sin  q  _  sin  ^(jo+g)  cosi(j>-g)  _  cosi(^-g)  .^. 

sin  (jo+g)     ~  &m\{p+q)  C0Bi{p+q)         cos^{p-\-q) 


sin;!? 
sin 


?  —  sin  g  _   sin  \{p—q)  cos-|(/>+g)  _  sin  \{p-q)     ^     .^. 
{P  +  Q)     ~  sm  i{p+q)  cos  i(p  +  q)         shii{p  +  q) 

sin  {p—q)     _  sin  UP~9)  C03i(p-g)  _  cos^(7>— ^7)  ,  ^  . 

sinp  — sinj'  ~  sin  ^(p— g-)  cos  ^(^+gj  "  cosiij)+q) 


all  of  wliich  give  proportiens  analogous  to  that  deduced  from 
Formula   ( 1 ). 

Since   the   second  members  of  (6)   and   (4)    are  the  same, 
we  have, 

sin  p  —  sm  q  _     sm  {p  -\-  q)    ^  ,     . 

sin  (p  —  q)      ~~  sin^  +  sin  2'  ' 


That  is,  the  sme  of  the  difference  of  two  arcs  is  to  the 
difference  of  the  sines  as  the  sum  of  the  sines  to  the  si)it 
of  the  sum. 

All  of  the  preceding  formulas  may  be  made  homogeneous 
in  terms  of  H,  Ji  being  any  radius,  as  explained  in  Art. 
30  ;  or,  we  may  simply  introduce  J?,  as  a  factor,  into  each 
term   as    many  times    as    may  be    necessary  to    render   all   of 


its  terms   of  the   same   degree. 


TRIGONOMETR  r.  71 

METHOD    OF    COMPUTING    A    TABLE    OF    NATURAL    SINES. 

68.  Since  the  length  of  the  semi-drcumference  of  a  circle 
whose  radius  Is  1,  is  equal  to  the  number  3.14159205  .  .  .  , 
f  we  divide  this  number  by  10800,  the  number  of  minutes 
n  180°,  the  quotient,  .0002908882...,  will  be  the  length 
)f  the  arc  of  one  minute ;  and  since  this  arc  is  so  small 
that  it  does  not  differ  materially  from  its  sine  or  tangent, 
this  may  be   placed   in   the   table   as   the  sine  of  one  minzite 

Formula   (3)    of    Table   11.,   gives, 

cos  1'   =    -y/l  —  siun'  z=   .9999999577     •     •     (l.) 

Having  thus  determmed,  to  a  near  degree  of  approxima- 
tion, the  sine  and  cosine  of  one  minute,  we  take  the  first 
formula   of  Art.    67,   and  put   it  under  the   form, 

sm  (a  +  b)    =    2  sin  a  cos  5  —  sm  (a  —  5), 

and   make   in   this,      b  =  1',      and  then  in   succession, 

a  =  1',  a  =  2\  a  =  3',  a  =  4',        &c., 

and   obtain, 

sin  2'  =  2  sm  1'  cos  1'  —  sin  0    =    .0005817764  .  .  . 

sin  3'  =  2  sin  2'  cos  1'  —  sin  1'  =    .0008726646  .  .  . 

sin  4'  =  2  sm  3'  cos  1'  —  sin  2'  =    .0011635526  .  .  . 

sin  5'  =  &c., 

thus    obtaining    the    sine    of    every    number    of    degrees    and 
minutes   from    1'    to    45°. 


72         ANALYTICAL      TRIGONOMETRY. 

The  cosines  of  the  corresponding  arcs  may  be  computed 
by   means   of   Equation    ( 1 ). 

Having  found  the  sines  and  cosines  of  arcs  less  than  45", 
those  of  the  arcs  between  45°  and  90°,  may  be  deduced, 
by  considering  that  the  sine  of  an  arc  is  equal  to  the  cosme 
of  its  complement,  and  the  cosine  equal  to  the  sine  of  the 
complement.      Thus, 

sm  50°  =  sin  (90°  —  40°)   =  cos  40°,         cos  50°  =  sm  40°, 

in  which  the   second   members    are    known  from  the   previous 
computations. 

To  i5nd  the  tangent  of  any  arc,  divide  its  sine  by  its 
cosine.  To  find  the  cotangent,  take  the  reciprocal  of  the 
corresponding   tangent. 

As  the  accuracy  of  the  calculation  of  the  sine  of  any  arc, 
by  the  above  method,  depends  upon  the  accuracy  of  each 
previous  calculation,  it  would  be  well  to  verify  the  work,  by 
calculating  the  sines  of  the  degrees  separately  (after  having 
found  the  sines  of  one  and  two  degrees),  by  the  last  pro- 
portion  of  Art.  67.       Thus, 

sin  1°     :     sin  2°  —  sm  1°     :  :     sm  2°  +  sm  1°    :     sin  3"  ; 
Bin  2*     :     sin  3°  —  ein  1°     :  :     sin  3°  +  sm  1°    :     sin  4°  ;  &c 


I 


SPHERICAL    TRIGONOMETRY. 


69.  SrnEKiCAL  Thigoxojietey  is  that  branch  of  Mathe- 
matics which   treats   of  the   solution   of  sj^herical   triangles. 

In  every  spherical  triangle  there  are  six  parts  :  three  sides 
and  three  angles.  In  general,  any  thi-ee  of  these  parts  being 
given,  the  remaming   parts  may  be  found. 

GENERAL    PRmCIPLES. 

70.  For  the  purpose  of  deducing  the  formulas  required 
in  the  solution  of  spherical  triangles,  we  shall  suppose  the 
triangles  to  be  situated  on  spheres  whose  radii  are  equal 
to  1.  The  formulas  thus  deduced  may  be  rendered  ai^plica- 
ble  to  triangles  lying  on  any  sphere,  by  making  them  homo- 
geneous in  terms  of  the  radius  of  that  sphere,  as  explained 
in  Art.  30.  The  only  cases  considered  will  be  those  in 
which  each  of  the  sides  and  angles  is  less  than   180°. 

Any  angle   of  a   spherical  triangle   is  the   same   as  the   dit>- 
dral   angle   included   by   the   planes   of  its   sides,    and   its   moa 
sure  is    equal    to    that    of    the    angle    included    betAvccn    two 
right    hues,    one    in    each    plane,   and    both    perpendicular    to 
their   common   intersection    at   the    same   point    (B.  VI.,   D.  4). 

The  radius  of  the  sphere  being  equal  to  1,  each  side  of 
the  triangle  will  measure  the  angle,  at  the  centre,  subtended 
by  it.       Thus,   in   the    triangle     ABC,     the   angle   at    A     \a 


SPHERICAL 


the   same   as    that    included    between   the   planes     AOC     and 

A  OB ;    and    the    side     a    is  the 

measure  of  the  plane  angle  BOC^ 

O      heinc:      the      centre     of     the 

pphere,  and    OB    the  radius,  equal 

to    1. 

71.  Spherical  triangles,  like 
plane  triangles,  are  divided  into 
two  classes,   right-anrjled  spherical 

triangles^   and    ohUque-angled  sjyherical    triangles.      Each   class 
will   he   considered   in    turn. 

We  shall,  as  before,  denote  the  angles  by  the  capital 
letters  A,  B,  and  C,  and  the  opposite  sides  by  the  small 
letters    a,    5,    and    c. 


FORMULAS     USED     IN     SOLVING     RIGHT-ANGLED     SPHERICAL 

TRIANGLES. 

72.  Let  CAB  be  a  spherical  triangle,  right-angled  at  A, 
and  let  0  be  the  centre  of  the 
sphere  on  which  it  is  situated. 
Denote  the  angles  of  the  triangle 
by  the  letters  A,  B,  and  C, 
and  the  opposite  sides  by  the 
letters  a,  b,  and  c,  recollecting 
that  B  and  0  may  change 
places,  provided  that  b  and  c 
change   places   at   the   same  time. 

DraAV  OA,  OB,  and  00,  each  of  which  will  be  equal 
to  1.  From  B,  draw  BP  perpendicular  to  OA,  and 
from  P  draw  PQ  perpendicular  to  OC  ;  then  join  the 
points  Q  and  B,  by  the  hne  QB.  The  line  QB  will  be 
perpendicular  to    OC    (B.  VI.,  P.  VL),   and   the   angle    PQB 


TRIGONOMETRY.  75 

will    be    equal    to    the    inclination   of   tlie    planes     0  GB    and 
OCA  ;    that   is,   it   Avill   he   equal   to   the   angle     C. 
We   have,    from   the   figure, 

PL  —  siu   c,      OP  =  cos  c,      QB  =  sin  a,      OQ  =  cos    a. 
Also,       j-j^  =  cos   f;        and       ^-5  =  sin  I. 

From  tlie  right-angled  triangles  OQP  and  QPB,  we  have, 
OQ  =  OP  cos  AOC;        or,        cos  a  =  cos  c  cos   5    .     (1.) 

PB  =  QB  sin   PQB;        or,        sin  c  =  sin   a  sin   C   .     (2.) 

OP 

Multiplying  both    terms  of   the  fi-action   ■—  by   OQ,   and 

remembering  that  cot  a  =  tan   (90°  —  a),  we  have, 

-^  =  -;-f,  X  ~;     or,    cos   C  =  tan    (90°  -  a)   tan   Z*.     (3.) 
IJlj         I^Jj         U(^ 

OP 

Multiply    both    terms    ol    the    fraction    j--j^,    by   PB,    and 

remembering  that  cot   C  =  tan    (90°  —  C),  we  have, 

OP        PB        OP  .     ,  .  ^^      ,    . 

■^yp   =  Yjp  X  Tw-,  ;       or,      sm  6»  =  tan  c   tan  (90°— C7).     (4.) 


If,   in     ( 2 ) ,    Ave    change    c    and     C,     into    h    and  J?,  we 
have, 

sin  5  =  sin  a    sin  J5 ( 5.) 

If,   in     ( 3 ),    we   change    h    and    C,     mto    c    and  By  we 
have, 

cos  B  =  tan  (90°  — a)  tan   c   •     •     •  •  (6, 

If^   in     ( 4 ),     we   change    b,    c,     and    C,     into    c,  5,  and 
5,    we  have, 

sin  c  =  tan  b   tan  (90°—^)     •     •     •  •  ( Y.) 


76  SPHERICAL 

Multiplymg    ( 4 )    by    ( 7  ),    member  by  member,  we  have,  \ 

sin  h    sin  c  =  tan  h    tan  c    tan  (90°— i?)  tan  (90°— C).  \ 

\ 

Dividing   both  members  by     tan  h  tan  c,       vre  have,  \ 

% 

;;. 

COS  h    cos  c  =  tan  (90°  — i?)  tan  (90°— (7)  ; 

and  substituting  for      cos  b  cos  c,      its  value,      cos  a,    taken 
from     ( 1 ),     we   have, 


cos 


a  =  tan  (90° -i?)  tan  (90°— (7J     •     •     (8.) 


Formula    ( 6 )     may  be   written   under   tbe   form, 


^         cos  rt    sm  c 

cos  Ji  =  -. 

sm  a    cos  c 


Substituting  for     cos  a,      its  value,     cos  b   cos  c,     taken   from 
(1  ),    and   reducing,   we   have, 

_         cos  b   sin  c 

cos    JJ    =    ; • 

sm  a 

Again,   substituting  for    sin  c,      its  value,     sin  a    sin  (7,     taken 
from    ( 2  ),     and  reducing,   we  have, 

cos  J^    =    cos  5    sin  C     •     •     •     •     (9.) 

Charging    .F,     5,     and     C,    in    (9 ),    ii:to    (7,    c,    and   U,     wc 

have, 

cos  C   =    cos  c    sin  ^   •     •     •     •     ( 10.) 

I 

These   ten    formulas    are    sufficient   for  the   solution   of  any 
right-angled  spherical  triangle  whatever. 


TRIGONOMETRY. 


77 


Napier's   ciucular  parts. 

73.  TIte  two  sides  ahout  the  right  angle,  c, 
the  compIeme)its  of  their  ojjposite  angles,  and 
the  comjHement  of  the  hgjJOthenuse,  are  called 
Napier'?  Circular  Parts. 

If  we  take  any  three  of  the  five  parts,  as 
shown  in  tlie  figure,  they  will  either  be 
adjacent  to  each  other,  or  one  of  them  will  be  separated  from 
each  of  the  other  two,  by  an  intervening  part.  lu  tlie  first  case, 
the  one.  lying  between  the  other  two  parts,  is  called  the  middle 
part,  and  the  other  two,  adjacent  parts.  In  the  second  case,  the 
one  separated  from  both  the  other  joarts,  is  called  the  middle  part, 
and  the  other  two,  opposite  parts.  Thus,  if  90°  —a,  is  the  middle 
part,  90°  —  B,  and  90°  —  C,  are  adjacent  jxtrls  ;  and  b  and  c,  are 
opposite  parts  J  and  similarly,  for  each  of  the  other  parts,  taken 
as  a  middle  part. 

74.  Let  us  now  consider,  in  succession,  each  of  the  five 
parts  as  a  middle  part,  when  the  other  two  parts  are  oppo- 
site. Beginning  with  the  hypothenuse,  we  have,  from  formulas 
(1),   (2),    (5),   (9),   and    (10),   Art.   72, 


sin   (90°-  a) 

sm  c 

ein  b 

sin  (90° -i?) 

sin  (90^— C) 


cos  b    cos  c (1.) 

cos  (90°— rt)  cos  (90°— C)     .  (2.) 

cos  (90°- a)  cos  (90°— .Z?)     •  (3.) 

cos  b    cos  (90°— (7)  ....  (4.) 

cos  c    cos  (90°— i>)  ....  (5.) 


Comparing  these  formulas  with   the   figure,   we  see  tliat. 

The  sine  of  the  middle  part  is  equal  to  the  rectangle  of 

the  cosines  of  the  opposite  parts, 

22 


78 


SPHERICAL 


Let  ITS  now  take  tlie  same  middle  parts,  and  the  other  parts 
adjacent.     Formulas   (8),   (7),   (4),  (G),   and  (3),  Art.  72,  give 


sin  (90°- a)    =  tan  (90°- i?)  tan    (90°- C) 

sin  c                  =  tan  h    tan  (90°—^)  • 

Bin  6                  =  tan  c    tan  (90°— C)  •     •     • 

sin  (90° -i?)    =  tan  (90°— a)  tan  c     •     •     • 

sin  (90°— C)   =  tan  (90°- a)  tan  5     •     •     • 


(6.) 

(^•) 
(8.) 

(9.) 

(10.) 


Comparing   these   formulas   with    the  figure,   we  see   that, 

The  swe  of  the  middle  2^'^'^'f  ^s  equcd  to  the  rectangle  of 
the   tangents   of  the  adjacent  parts. 

These  two  I'ules  are  cahed  Napier's  rules  for  Circular 
Parts,  and  they  are  sufficient  to  solve  any  right-angled 
spherical   triangle. 

75.  In  applying  Napier's  niles  for  circular  parts,  the  part 
sought  will  he  determined  hy  its  sine.  Now,  the  same  sine 
corresponds  to  two  different  arcs,  supplements  of  each  other  ; 
it  is,  therefore,  necessary  to  discover  such  relations  between 
the  given  and  required  parts,  as  will  serve  to  point  out 
which    of  the   two    arcs  is   to   he   taken. 

Two  parts  of  a  spherical  triangle  are  said  to  he  of  the 
ame  species,  when  they  are  both  less  than  90°,  or  holli 
greatei  than  90°  ;  and  of  different  species,  when  one  is  less 
and   the   other  greater   than    90°. 

From   Formulas    (9)    and    (10),    Art.   72,   we   hav<^. 


sin  C  = 


cos  J^ 
cos  b  ' 


and       sin  S 


cos  C 
cos  e 


TllIGONOMETUY.  Tli 

since  the  angles  B  and  C  are  lotli  less  than  180°,  their 
sines  must  always  be  positive  :  hence,  cos  B  must  have 
the  same  sign  as  cos  J,  and  the  cos  C  must  have  the 
same  sign  as  cos  c.  This  can  only  be  the  case  when  B 
is  of  the  same  species  as  h,  and  C  of  the  same  speciea 
as  c  ;  that  is,  the  sides  about  the  right  angle  are  always 
of  the  same  species   as   their  opposite   angles. 

From  Formula  ( 1 ),  we  see  that  when  a  is  less  than 
90°,  or  when  cos  a  is  positive,  the  cosines  of  b  and  c 
^vill  have  the  same  sign  ;  that  is,  b  and  c  will  be  of  the 
same  species.  When  a  is  greater  than  90°,  or  when  cos  a 
is  negative,  the  cosines  of  b  and  c  will  be  contrary ;  that 
is,  b  and  c  will  be  of  different  species :  hence,  when  the 
hypothenuse  is  less  than  90°,  the  txoo  sides  about  the  right 
angle,  and  consequently  the  two  oblique  ajigles,  toill  be  of  the 
same  species  ;  when  the  hypothenuse  is  greater  than  90", 
the  two  sides  about  the  right  angle,  and  consequently  the  two 
oblique   angles,  iciU  be   of  different  species. 

These  two  principles  enable  us  to  determine  the  nature 
of  the  part  sought,  in  every  case,  except  when  an  oblique 
angle  and  the  opposite  side  are  given,  to  find  the  remaining 
parts.  In  this  case,  there  may  be  tioo  solutions,  one  solu- 
tion,  or  no  solution   at  all. 

Let    BAC    be   a  right-an- 
gled   triangle,     in     which     ^       ^X  j  \  ^g, 
and     b     are    given.       Prolong 
the  sides    BA     and    BC    till 
ihey     meet     in      B'.        Take 

B'A'  =  BA,  B'C  =  BC,  and  join  A'  and  C  by  the 
arc  of  a  great  circle  :  then,  because  the  triangles  BAC  and 
B'A'C  have  two  sides  and  the  included  angle  of  the  one, 
equal  to  two  sides  ?.nd  the  included  angle  of  the  other,  each 
to   each,   the   remainhig   parts    will    be    equal,    eacl:     to    each : 


80 


SPIIEPwICAL 


that  is,  A'C  =  AC,  and  the  angle  A  equal  to  the 
ungle  A  :  hence,  the  two  triangles  BACy  jB'A  C\  are 
right-angled  ;  they  .  have  also 
one  oblique  angle  and  the  op- 
gKidte  side,  in  each,  equal. 

Now,  if  b  differs  inore   from 
.10°  than  J5,  there  will   evident- 
ly  he   two   solutions,   the    sides 
includiug   tlie.  given   angle,    in    the   one   case,  being   supplements 
of  those  which  include  the  given  angle,  in  the  other  case. 

If   h  =  JBy    the    triangle    will    be    bi-rectangular,    and    there 
will  be  but  a  single  solution. 

If  b  differs  less  from  00°  than  H,  the  triangle  cannot  be  con- 
structed,  that   is,   there   Avill   be   no   solution. 


SOLUTION     OF    EIGHT- ANGLED     SrHEKICAL    TKI  ANGLES. 

76.  In  a  right-angled  spherical  triangle,  the  right  angle 
13  always  known.  If  any  two  of  the  other  parts  are  given, 
the  remaining  parts  may  be  found  by  Napier's  rules  for 
circular   parts.       Six   cases    may   arise.       There   may   be   given, 

I.  The  hypothenuse   and    one   side. 

II.  The  hypotheuuse   and   one    oblique   angle. 

III.  The  two   sides   about   the   right   angle. 

IV.  One  side   and   its   adjacent   angle. 
V,  One  side   and    its   opposite    angle. 

VI.    The   two   oblique   angles. 

In  any  one  of  these  cases,  we  select  that  part  which  is 
eflher  adjacent  to,  or  separated  from,  each  of  the  other  given 
parts,  and  calling  it  the  middle  part,  we  employ  that  one  of 
Napier's  rules  which  is  applicable.  Having  determined  a  third 
part,   the   other  two   may  then  be  found  in   a  sunilar  manner. 


TRIGONOMETRY. 


81 


It  is  to  be  observed,  that  the  formnhxs  employed  are  to  be 
rendered  homogeneous,  in  terms  of  R,  as  explained  in  Art.  30. 
This  is  done  by  simply  multiplying  the  radius  of  the  Tables, 
R,   into   the  middle  part. 


EXAMPLES. 

1.    Given   a   =    105°   17'  29",   and 
47'    11",     to     find    C,     c, 
and    B. 


d    =    38° 


Since  a  >  90°,  b  and  c  mnst  be 
of  different  species,  that  is,  c  >  90° ; 
for   the    same    reason,  C  >  90°. 


OPERATIO]Sr. 

Pormnla   (10),   Art.   74,   gives,   for   90°  -  C,   middle  part, 

log  cos   C  =  log  cot  a  +  log  tan   5  —  10 ; 
log  cot  a    (105°  17'  29")     9.430811 
log  tan  b    (  38°  47'  11")     9.905055 

log  cos  6' 9.341SGG  .-.  C=  102°  41'  33" 

Formula    ( 2 ),   Art.    74,   gives   for   c,   middle   part, 
log   sin   c  =  log   sin  a  +  log   sin    C  —  10 ; 


log  sin  a     (105°  17'  29")     9.98434G 
log  sin  C    (102°  41'  33")     9.989256 

log  sin  c 9.973G02  .  • .   c  =  109°  4G'  32". 


Formula    (4),   gives,   for   90°  —  B,   middle   part, 

s 

log  COS  B  =  log  sin   C  +  log  cos  5  —  10 ; 


log  sin  C    (102°  41'  33")     9.989256 
log  cos  b     (  38°  47'  11")     9.891808 


loff  cos  B 


9.8810G4 


B  =  40°  29'  50". 


Ans.    c  =  109°  4G'  32",     B  =  40°  29'  50",     C  =  102°  41'  33". 


82  •        SPHERICAL 

2.     Given    I    =    51°    30',    nud    B   =  58°    35',    to    find    c, 
a,    and    C. 

Because   b  <.  B,   there  are  two   tulutious. 

OPERATIOi^. 

Formula   ( 7 ),  gives  for  c,   middle  part, 

log   sin   c  —  log  tan    b  +  log   cot   ^  —  10; 

log  tan  b      (51°  30')     .     10.099395 
log  cot  B     (58°  35')     .       9.785900 

log  sin  c      ....       9.885295     .-.     c  =     50"  09'  51", 

and    c  =  129°  50'  09". 

Formula   ( 1 ),  gives  for   90°  —  a,   middle  part, 

log  cos  a  =  log  cos   b  +  log  cos  c  —  10 ; 

log  cos  b     (51°  30')       .      9.794150 
log  cos  c     (50°  09'  51")      9.80G580 

log  cos  a      ....     9.600730    .-.    a  =    6G°  29'  54", 

and  a  =  113°  30'  06". 

Formula   ( 10 ),  gives  for  90°  —  C,  middle  part, 

log  cos  C  =  log  tan  b  +  log  cot  a  —  10  ; 

log  tan  b     (51°  30')     •      10.099395 
log  cot  a      (CG°  29'  54")     0.638336 

log  cos  C      ....     9.737731    .' .    C  =     56°  51'  38", 

and     C  =   123°  08'  22". 
In   a  Biniilar  manner,    aU   other   cases  may   be  solved. 

3.     Given     a  =  86°  51',      and     B  =  18°  03'  32",     to   find 
6,     c,     and     C. 

Ans.     b  =  18°  01'  50",      c  =  86°  41'  14",     C  =  88°  58'  25". 


TRIGONOMETRY.  83 

4.  Given      h  -   155°  2V'  54",      and      c   —  29°  4C'  08",     to 
find      a,      -C,      and      C. 

Ans.     a  =  142°  09'  13",     J3  =  137°  24'  21",      C  =  54°  01'  16". 

5,  Given      c  =   73°  41' 35",     and      J?  =  99°  17' 33",      to 

find      a,      6,      and     C. 

Ans.     a  r=  92°  42'  17",     b  =  99°  40'  30",     O  =  73°  54'  47". 


6.  Given      b  =   115°  20',     and     B  =  91°  01'  47",     to  find 
a,      c,      and      C. 

(    64°  41'  11",  r  177°  49' 27",  f  177°  35' 36". 

a  =  -l  c  =  -{  C  =  i 

[  115°  18' 49",  [      2°  10' 33",  [      2°  24' 24". 

7.  Given     i?  =  47°  13'  43",      and      C  =  126°  40'  24",      to 
find      a,      b,      and      c. 

Ans.     a  =  133°  32'  26',     b  =  32°  08'  56",     c  =  144°  27'  03". 


In  certain  cases,  it  may  be  necessary  to  find  but  a  single 
part.  This  may  be  effected,  either  by  one  of  the  formulas 
given  in  Art.  74,  or  by  a  slight  transformation  of  one  of 
them. 

Thus,  let  a  and  B  be  given,  to  find  C.  Regarding 
90°  —  a,     as  a   middle   part,    Ave   have. 


vr^ence, 


cos  a    =    cot  J3   cot  C  ; 

^         cos  a 

cot  C    =    =s  > 

cot  J?  ' 

and,   by   the   application   of  logarithms, 

log  cos  a  +  (a.  c.)   log  cot   B  =  log  cot   C; 

from   which    C  niav   be   found.      In    like   manner,   other   cases 
may   be    treated. 


u 


SPHERICAL 


QUADRANTAL     SPHERICAL    TRIANGLES. 

77.  A  QuADRANTAL  SpiiERiCAL  Triangle  is  One  in  wbicL 
one  side  is  equal  to  90°.  To  solve  such  a  triangle,  -vve  pass 
to  its  polar  triangle,  by  subtracting  each  side  and  each 
angle  from  180°  (B.  IX.,  P.  VL).  The  resulting  polar  tri- 
angle Avill  be  right-angled,  and  may  be  solved  by  the  rules 
already  given.  The  polar  triangle  of  any  quadrantal  triangle 
being  solved,  the  parts  of  the  given  triangle  may  be  found 
by  subtracting   each   part   of  the   polar   triangle   from    180°. 

EXAMPLE. 


Let  A'B'C  be  a  quadrantal 
triangle,  in  which  B'C  =  90  ^ 
J3'  =   75°  42',      and     c'   =   18°  37'. 

Passing  to  the  polar  triangle, 
we   have, 


A  =  90°,       b  =   104°  18',       and       C  =   161°  23'. 

Solving   this   triangle    by   previous   rules,    we    find, 
a  =   76°  25'  11",  c  =    161°  55'  20",  i?  =   94°  31'  21"  ; 

bence,  the  required  parts  of  the  given  quadrantal  triangle  are, 
A'  -   103°  34'  49"  C  -   18°  04'  40",         b'  =   85°  28'  39". 

In   a    similar   manner,    other    quadrantal    triangles    may   he 
solved. 


.^ 


TRIGONOMETRY. 


85 


FORMULAS    USED    IN     SOLVING     OBLIQUE- ANGLKD     81'IIKRICAL     TRI- 
ANGLES. 

78.  Let  AUG  represent  an  oblique-angled  spherical  tri 
Siigle.  From  either  vertex,  C, 
draw  the  arc  of  a  great  circle 
CIi\  perpendicular  to  the  oppo- 
site side.  The  two  triangles 
ACJJ'  and  r>CJi'  wiU  be  right- 
angled    at     i>'. 

From  the  triangle  A  CB\    we 
have   Formula   ( 2  ),    Art.  74, 

sin  C12'  =   sin  A  sui  b. 


From   the   triangle    JjCB\     we  have, 

sin  CB'  =   sin  B  sin  a. 
Equating  these   values  of     sin  CB\      Ave   have, 
sin  ^  sin  6    =    sin  i>    sin  a  ; 

from  which  results   the  proportion, 

sin  a    :     sm  5    :  ;     sin  J.     :     sin  7?    .     .     .     (l.) 


In  like  manner,  we  may  deduce, 
sin  a  :  sin  c  :  ;  sin  A 
sin  h     :     sin  c    :  :     sin  B 


sin  6'    .     .     .     ( 2.) 
sin  C    .     .     .     (3.) 


That    is,    in    any    spherical    triangle,    the    sines    of    the    sid<. 
are  jyroportional  to   the  sines   of  tJieir  opposite  angles. 

Had  the   perpendicular   fallen   on   the  prolongation   of   AB, 
the   same   relation    would   have  been   found. 


8G 


S  P  H  E II I  C  A  L 


79.     Let    ABC    represent    any    splierical    triangle,   and     0 
the     centre     of    the     sphere      on 
which    it    is    situated.      Draw   the 
radii     OA,     OB,     and     0C\    from 
;^      draw      CP      perpendicular    to 
,lie   plane    AOB  \    from     P,      the 
foot    of    this    perpendicular,    draw 
PD      and     BE     respectively   per- 
pendicular to    OA    and   OB  ;    join 
CD     and     CE,     these   lines   will  be   respectively   perpendicular 
to     OA      and     OB     (B.  VI.,   P.  VI.),   and   the   angles     CBP 
and     CEP    will   be   equal  to   the   angles    A     and    B    respec- 
tively.     Draw    BL    and    PQ,    the  one  perpendicular,   and   the 
other   parallel   to     OB.      "We   then   have, 


OE  =  cos  a,        DC  =.  sin  5, 
We   have   from  the   figure, 

OE  =   OL  +  QP 


OD 


cos  h. 


(1.) 


In   the   light-angled    triangle     OLD, 

OL  =    OD    cos  DOL  =  cos  b   cos  c. 

The  right-angled  triangle  PQD  has  its  sides  respectively 
perpendicular  to  those  of  ODD ;  it  is,  therefore,  similar  to 
it,   and   the   angle    QDP    is   equal   to     c,     and   we   have, 


QP  =  PD    sin  QDP  =  PD  sin  c    •     • 

rhe   right-angled   triangle     CPD     gives, 

PD  =    CD  cos  CDP  =  sin  5   cos  ^  ; 

substituting   this   value   in    ( 2 ),    we   have, 

QP  =  sin  5   sin  c   cos  A  ; 


(2.) 


TRIGONOMETRY.  87 

and  now  substituting  these  values  of  OE^  OL,  and  QP^ 
in    ( 1 ),    we   have, 

cos  a  =  cos  h   cos  c  +  sm  J   sin  c  cos  A  •     (3.) 

Id    the   same   way,    we  may   deduce, 

cos  b  =.  cos  a  cos  c  +  sin  a   sin  c   cos  jB     •    •     (4.) 
cos  c  =   cos  a   cos  J  +  sin  a   sin  h   cos  C     •     •     (5.) 

That  is,  the  cosine  of  either  side  of  a  spherical  triangle  is 
equal  to  the  rectangle  of  the  cosines  of  the  other  two  sides 
plus  the  rectangle  of  the  sines  of  these  sides  into  the  cosine 
of  their  included  angle. 

80.  If  we  represent  the  angles  of  the  polar  triangle  of 
ABC,  by  A\  B\  and  C",  and  the  sides  by  a\  h' 
and    c',     we  have   (B.  IX.,  P.  VI.), 

a  =  180°  -  A',       b  =   180°  -  B\      c  =  180°  —  C", 
A  =   180°  -  a',      B  =  180°  -  b\       C  =  180°  -  c'. 

Substituting  these  values  in  Equation  (3),  of  the  preceding 
article,    and   recollecting   that, 

cos  (180°-^')  =  -  cos  A',       sin  (180°- i?')  =  sin  B',    &c., 
we   have, 

—  cos  A'  =  cos  B'   cos  C  —  sin  B'  sin  C   cos  a'  ; 

or,  changing  the  signs  and  omitting  the  primes  (since  the 
preceding   result   is   true   for   any   triangle), 

cos  A  =  sin  B   sin  C  cos  a  —  cos  B   cos  C         ( 1.) 


88  SPHERICAL 

In   the   same   "way,   we   may   deduce, 

cos  B  =   Bin  A   6m  C  cos  J  —  cos  A   cos  C    •     (2.) 

cos  G    =    sin  J.   sin  jB   cos  c  —  cos  -4   cos  B    -     (3.) 

Tluit  is,  the  cosine  of  either  angle  of  a  spherical  triangh 
is  equal  to  the  rectangle  of  the  sines  of  the  other  tioo 
angles  into  the  cosine  of  their  included  side^  minus  tin 
rectangle  of  the  cosines   of  these    angles. 

81.     From  Equation    (8),    Art.  T9,   we   deduce, 

.         cos  a  —  cos  5   cos  c  ,    ^ 

cos  A     =    : 1 r i.1-; 

sin  0   sin  c 

If  we  add  this  equation,  member  by  member,  to  the  num. 
ber  1,  and  recollect  that  1  +  cos  A,  in  the  first  membetj 
is   equal   to      2  cos^^     (Art.  66),   and   reduce,    we   have, 

sin  b   sin  c  +  cos  a  —  cos  h   cos  c 

2  0,0'^  \A   =   : — 7 — —' > 

■'  sm  o   sui  c 

or,    Formula    ( © ),    Art.  65, 

2  e„#J4   =  !^_^J^(i±l)        (2.) 

''  sm  0    sm  c 

And   since,   Formula   (SI),    Art.  67, 

cos  a  —  cos  {b  +  c)   —  2  sin  \{a -\- b  ^-  c)  sin  ^(5  +  c  —  a), 

Equation    (2)    becomes,   after   dividing    both    members  by    2 

„  ,  ^         sin  Ua  +  J  +  c)  sin  \(b  -{-  c  —  a) 

cos^  iA  =  ^'^ -. — 7 — :- * 

■*  sin  6   sm  c 


TRIGONOMETRY.  8J) 

If,   in   this   we   make, 

^{a  -{-  b  -{-  c)  =  Is  ;  whence,  ^(5  -[-  c  —  a)  ==  |s  —  a, 

ard    extract   the   square   root   of  both   members,   we   have, 

,    >  /  sin  ^s    sin  Us  —  a)  ,     ^ 

cos  ^A    =    \/  -.—T~^- •    •    •     •     (3.) 

V  sm  o  sm  c  ' 

That  is,  the  cosine  of  one-half  of  either  amjle  of  a  spherical 
triangle^  is  equal  to  the  square  root  of  the  sine  of  one-half 
of  the  sum  of  the  three  sides,  into  the  sine  of  one-half  this 
sum  minus  the  side  opposite  the  angle,  divided  by  the  rect- 
angle  of  the  sines   of  the   adjacent   sides. 

If    we    subtract    Equation    (  1 ),     of    the    preceding  article, 
member   by   member,   from   the   number    1,    and   recollect  that, 

1  —  cos  yl    =    2  siu2  ^A, 
we   find,    after   reduction, 


.      .    -               /  sm  (hs  —  h)  sm  {^s  —  c)  ,  ,  . 

&in  \A    =    \    ^- — : — — .     .     .     (4.) 

V  sm  0  sm  c 

Dividing   the   preceding    vaUie  of      sin  ^  A,      by      cos  \  A, 
we   obtain, 

,    .                /  sin  ils  —  b)  ein  (is  —  c)  ,     ^ 

V  sm  \s  sm  \\s  —  a)  ^     ' 


82.     If    the    angles    and    sides    of    the    polar    triangle    of 
ABC    be   rei^resented   as   in   Art.  80,   we   have, 

A  =  180=  —  a',         b  =   180° -i?',  c  =   180°  — C", 

ke  =  2V0=  -  i{A'+Ii'-\-  C),       \a-a  =  90°-i(Z?'  f-  C'-A'). 


90  SPHERICAL 

Substituting   these   values  in    (3),    Art.    81,    and    reducing 
by  the   aid  of  the   formulas  in  Table   m.,   Art.  63,   we   find, 


sin  ^a' 


/-  cos  UA'+J3'+  C)    cos  }':B'^  C'-A') 
-    V  sin  B'  sin  G' 

Placing 

^{A'+Ii'-\-C')  =  iS;      whence,      ^B'+C'-A')  =  ^S-A'. 

Substituting   and   omitting  the   primes,   we  have, 


.     ,  /  —  cos  ^S  cos  {^S  —  A)  .     . 

In   a  similar   way,   we   may  deduce   from    (4),   Art.  81. 

/cos{^S-B)  cos  {jS-C)  ,  ^  . 

cos  Aa    =    \/ . — ^ — -. — y^ •    •     K-^-) 

^  V  sm  i>  sm  C 


and  thence. 


/—  cos  ^S   cos  {^S  —  A)  I  -  > 

tan  ia    =    Vcosa>S-i?)cosa^-C^)  '     '     *     ^^'^ 

83.     From  Equation    ( 1 ),   Art.  80,    we   have, 

.     -^   .    ^  .    _,  sin>4    .    , 

cos  A  +  cos  jB  cos  C  =  sm  ^  sm  (7  cos  a  =  sm  C  -. sm  6  cos  a  ; 

sm  a 

(1.) 

since,   from   Proportion    ( 1 ),   Art.  78,     we  have, 

.     „  sin  ^         , 

sm  X?    = sm  0. 

sm  a 

Also,   from   Equation    (2),   Art.  80,    we  have, 

KIT!    y4 

cos  ^  -f  cos  A  cos  C  =  sin  ^  sin  C  cos  6  —  sin  (7    .         sin  a  cos  h 
'  sm  a 

(2.) 


TRIGONOMETRY.  91 

Adding    ( 1 )    and    ( 2 ),   and   dividing  by     sin  C,     we   obtain, 

/        A    I  jy.   I  +  cos  0  smA 

(cos  A  +  cos  JJ)  ; — .^ —    =    -. sin  (a  -\-  o).     (  3.) 

^  '^        sm  C  sm  a  /       \     / 

The   proj^ortion,  sin  A     :     sin  ^    :  :     sin  a    :     sin  5, 

taken    first  by  composition,    and   then   by    division,   gives, 

sin  yl  +  sin  i>    =    -; (sin  a  +  sin  5)  •     •     •     ( 4.) 

sm  «    ^  '  ^     ' 

gin  ^  —  sin  ^    =    -. (sin  a  —  sin  J)  •     •     •     (  5.) 

sm  a    ^  ^  ^     ' 

Dividing    ( 4 )    and    ( 5  ),   in   succession,   by   ( 3 ),  we  obtain, 

.  sin  ^  +  sin  i?  sin  G       _    sin  nr  +  sin  5 

cos  A  +  cos  B        1  +  cos  C   ~      sin  {a  +  d) 

sin  ^  —  sin  ^  sin  (7        _    sin  «  —  sin  J 

cos  A  +  cos  B        1  +  cos  C   ~      sin  (a  -{-  b)  ^    '^ 

But,  by  Formulas   (2)    and    (4),  Art.  67,   and  Formula   (SJ")» 
Art,  66,   Equation   ( 6  )    becomes, 

w  J        T^x                   1  ^  cos  A(«  —  b)  ,     . 

tan  i{A  +  B)    z=    cot  ^C  ,    ,     ;     •     •     (8.) 

^  ^  cos  y(«  -h  b)  ^     ' 

and,    by   the    similar     Formulas   ( 3 )    and    ( 5 ),      of    Art.    67, 
Equation    (  7 )    becomes, 

, ,  A        -r»x  ,  y-r  sin  i(a  —  b)  ,     - 

tan  U^  -  JS)    =    cot  \  C  -r--r)——j^  •      .    .     (  9.) 
^  ^  '      sm  j{a  -[-  b)  ^ 

These  last    two    formulas  give    the    proportions   known   ae  the 
first  set  of  Napier'^s  Ajialogies. 

co6^(a  +  J)    :    cos|(«— J)     :  :    cot^C    :    tan^(^  +  i?).     (10.) 

sin  ^-(a  +  i)     :     sin  |(a— J)     :  :     cot^(7    :    iaxi^{A—B).     (11-) 


92  SPHERICAL 

If  in  these  we  substitute  the  values  of  a,  b,  C,  A^ 
aud  Jj,  in  teniis  of  the  corrcspondiug  parts  of  the  polar 
triangle,    as   expressed   in   Art.  80,  Ave   obtain, 

QOsi{A  +  Ij)  :  cosl{A  —  U)  :  :  tan^c  :  tan^(a  +  i).  (12.) 
&ini{A-{-I])  :  Bmi(A~JJ)  ::  tan  ^-c  :  iani{a-b).  (13.) 
the  second  set  of  I^^apier^s  Analogies. 

In  applying  logaritlims  to  any  of  the  preceding  formulas, 
they  must  be  made  homogeneous,  in  terms  of  72,  as  ex- 
plained  in   Art.   30. 

SOLUTION    OF    OBLIQUE-ANGLED    SPHERICAL    TRIANGLES. 

84.  In  the  solution  of  oblique-angled  triangles  six  differ- 
ent  cases  may  arise  :    viz.,   there   may  be   given, 

I.  Two   sides   and  an   angle   opposite   one  of  them. 

n.  Two    angles   and   a   side   opposite   one   of  them. 

m.  Two   sides   and   their  included   angle. 

IV.  Two   anijles   and   their  included   side. 

V.  The   three   sides. 

YI.  The   three   angles. 

CASE      I. 

Given   two   sides   and  an  angle  opposite  07ie  of  them. 

85.  The  solution,  in  this  case,  is  commenced  by  finding 
liie  angle  opposite  the  second  given  side,  for  which  purpose 
Formula   ( 1 ),   Ait.  T8,    is   employed. 

As  this  angle  is  found  by  means  of  its  sine,  and  because 
the  same  sine  corresponds  to  two  different  arcs,  there  would 
eeem  to  be  two  different  solutions.  To  ascertain  when  there 
are  two  solutions,  when  one  solution.,  and  when  7io  solution 
at  all,   it   becomes   necessary   to    examine    the    relations  which 


I 


TRIGONOMETRY. 


93 


may  exist   between    the    given    parts.       Two   cases   mny   arise, 
viz.,   the  given   angle   may  be   acute^   or   it   may  be   obtuse. 

We  shall  consider  each  case  separately  (B.  IX.,  P.  XIX.. 
Gen.  Scholium). 

First  Case.  Let  A  be 
the  given  angle,  and  let  a 
and  b  be  the  given  sides. 
Prolong  the  arcs  A  C  and 
AB  till  they  meet  at  A\ 
forming  the  lune  AA' )  and 
from  (7,  draw  the  arc  CB'  perpendicular  to  ABA'.  From 
C,  as  a  pole,  and  with  the  arc  a,  describe  the  arc  of  a 
small  circle  BB.  If  this  circle  cuts  ABA\  in  two  points 
between  A  and  A\  there  will  be  two  solutio7is ;  for  if 
C  be  joined  with  each  point  of  intersection  by  the  arc  of 
a  great  circle,  we  shall  have  two  triangles  ABC^  both  of 
which   will   conform   to   the   conditions   of  the   problem. 

If  only  one  point  of  in- 
tersection lies  between  A 
and  A\  or  if  the  small 
circle  is  tangent  to  ABA\ 
there  will  be  but  one  solu- 
tion. 

If  there  is  no  point  of  intersection,  or  if  there  are  points 
of  intersection  which  do  not  lie  between  A  and  A\  there 
nvill  be   no   solution. 


From  Formula   (2),  Art.  V2,   we  have, 

sin  CB'  =    sin  b   sin  A^ 

from  which    the    perpendicular,   which   wiU  be   less    than     90°, 

will    be    found.       Denote   its  value  by   p.      By  inspection  of 

the  figure,   we   find  the   follo-\ving   relations  : 

23 


94 


SPHERICAL. 


1.  When  a  is  greater  than  p,  cmd  at  the  same  time  less 
than  both   b   and  180°  —  b,    there   will  he   tico  solutions. 

2.  When  a  is  greater  than  p,  and  intermedlaie  in  value 
between  b  ajid  180°  — b;  or,  when  a  is  equal  to  p,  then 
will  he   but   one  solution. 

If  a  =  b,  and  is  also  less  than  180°  —  5,  one  of  the  poiiila 
o!"  intei-section  will  be  at  A,  and  there  will  be  but  one 
solution. 

3.  When      a     is  greater  than     p,     and  at  the  same  time 


a 


greater   than    both',    b      and      180°  —  b  ;      or,   tchen 
less  than      p,      there  will  be  no   solution. 


Second  Case.  Adopt  the 
same  construction  as  before. 
In  this  case,  the  perpendicu- 
lar will  be  greater  than  90°, 
and  greater  also  than  any 
other  arc  CA,  GB,  CA\ 
that   can    be   drawn   from     C 

to    ABA'.       By   a   course   of  reasoning   entirely  analogous    to 
that  in   the  preceding  case,  we  have   the  following  principles: 


4.  When      a 
greater  tliati   both 
solutions. 

5.  Whe7i      a 
value  between      b 


is  less  than      p,       and  at    the  same    time 
b      and      180°  —  b,       there  will    be    two 


is    less    than 


and    intermediate    in 


and      180°  —  b  ;      or,  when      a      is  equal 
to      p,      there  will    be  but   otie   solution. 

6.      When      a      is   less    than      p,      a}id  at   the   same   time 


less    than    both 


and       ISO" 


b  ; 


or. 


wJien 


ta 


greater   than      p,       there  will  be  no   solution. 


Having  found  the  angle  or  angles  opposite  the  second 
side,  the  solution  may  be  completed  by  means  of  Napier's 
Analogies. 


I;     « 


i 


TRIGONOMETRY.  95 

EXAMPLES. 

1.  Given  a  =  43^  27'  36",  h  =  82°  58'  17",  and 
A   =  29°  32'  29",       to   find      B,      C,      and      c. 

We  see  at  a  glance,  that  a  >  j^,  since  p  cannot 
exceed  A  ;  we  see  further,  that  a  is  less  than  both  b 
and  180°  —  b  ;  hence,  from  the  first  condition  there  will  be 
two   sokitions. 

Applying   logarithms  to   Formula  ( 1 ),   Art.    78,    we    have, 

[SL.  c.)   log  sin  a  +  log   sin  0  +  lug  sin   yl  —  10  =  log  sin  B ; 

(a.  c.)  log  sin  a    .    .  (43°  27'  36")     .    .     .    0.1G2508 

log  sin  5    .    .  (82°  58'  17")     .    .    .    9.996724 

log  sin  A   .    .  (29°  32'  29")     .    .    .     9.692893 

log  sin  i? 9.852125 

.-.  i?  =  45°  21'  01",     and  B  =  134°  38'  59". 

From   the  first  of  Napier's  Analogies   ( 10 ),   Art.  83,  we  find, 

(a.  c.)  log  cos  ^  {(i—h)  +  log  cos  -I  {a+l)  +  log  tan  ^  (^  +^)  — 10 

=  log  cot  i  C. 

Taking  the  first  value  of  B,  we  have, 

\{A  -\-  B)  =  37°  26'  45" ; 
also, 

\{a  +  h)  =  63°  12'  56" ;         and,         ^  (a  -  J)  =  19°  45'  20". 

(a.  c.)  log  cos  I  [a  -I)       .     (19°  45'  20")     .  0.026344 

log  cos  l{a  +  b)       .     (63°  12'  56")     .  9.653825 

log  tan  i{A  +  B)    .     (37°  26'  45")     .  9.884130 

log  cot  ^  C     . 9.564299 

.-.    i  C=  69°  51'  45",      and     0=  139°  43'  30". 


96  SniERlCAL 

The    side    c  muy    be    found    by    means    of    Formala    (12), 
Art.   83,   or   by   meiins   of  Formula   (2),   Art.  78. 
Applying   logarithms   to   the   proportion, 

siu   A     :     sin  C    :  :     sin  a    :     sin   c,        we  have, 

(a.  c)  log  sin  A  =  log  sin   C  +  log  sin  a  —  10  =  log  sin   c ; 

(a.  c.)  log  sin  A     (  29°  32'  29")  0.307107 

log  sin  G    (139°  43'  30")  9.810539 

log  siu  a     (  43°  27'  3G")  9.837492 

log  sin  c 9^95513_8  .-.  c  =  115°  35'  48". 

We  take  the  greater  value  of  c,  because  the  angle  (7, 
being  greater  than  the  angle  i?,  requires  that  the  side  c 
should  be  greater  than  the  side  b.  By  using  the  second 
value   of    J3,     we   may  find,   in   a   similar   manner, 

C   =    32°  20'  28",  and  c    =    48°  16'  18". 

2.  Given  a  =  97°  35',  b  =  27°  OS'  22",  and 
A   =  40°  51'  18",         to   find      i>,      C,      and      c. 

Ans.     J5  =  1V°31' 09",    (7  =  144°  48' 10",     c  =  119°  08' 25". 

3.  Given  a  =  115°  20'  10",  5  =  57°  30'  06",  and 
A  =  126°  37'  30",      to   find      Jj,     C,      and      c. 

Ans.     B  =  48°  29'  48",      C  =  61°  40'  16",     c    =  82'"'  34'  04". 

CASE      n. 

Given   two   angles   and  a  side  opposite   07ie  of   them. 

86.  The  solution,  in  this  case,  is  commenced  "by  finding 
the  side  opposite  the  second  given  angle,  by  means  of  For- 
mula ( 1  ),  -Art.  78.       The   solution   is   completed  as  in  Case  L 


TRIGONOMETRY. 


97 


Since  the  second  side  is  found  by  means  of  its  sine,  there 
may  bo  two  solutions.  To  investigate  this  case,  i^e  pass  to 
the  polar  triangle,  by  substituting  for  each  part  its  supple- 
ment. In  this  triangle,  there  will  be  given  two  sides  and 
an  angle  opposite  one  ;  it  may  therefore  be  discussed  as  in 
the  preceding  case.  When  the  polar  triangle  has  two  sohir 
tions^  one  solution,  or  no  solution,  the  given  triangle  will, 
in  like  manner,  have  two  solutions,  one  solution,  or  no  solnr 
tion. 

The  conditions  may  be  written  out  from  those  of  the  pre- 
ceding case,  by  simply  changing  anf/les  into  sides,  and  the 
reverse  ;    and   greater  into   less,   and   the  reverse. 


Let  the  given  parts  be  A,  J3, 
and  a,  and  let  ^>  be  an  arc 
computed   from   the   equation, 


sin  p    =    sin  a    sin  H, 

There  will   be   two  cases  :      a      may  be  greater  than   90"  ; 
or,      a      may   be   less    than   90°. 


In   the   first   case, 

1.  When  A  is  less  than  p,  a7id  at  the  same  time 
greater  than  both  B  and  180°  —  B,  there  loill  be  tvHo 
solutions. 

2.  When  A  is  less  than  p,  and  intermediate  in 
value  bettceen  B  and  180°  —  i?  /  or,  when  A  is  equal 
*o    p,      there  will  be  but  one  solution. 

3.  When  A  is  less  than  p,  and  at  the  same  time 
less  than  both  B  a?id  180°  -  -B  /  or,  when  A  is 
greater  than     p,      there  will  be  no   solution. 


98 


SPHERICAL 


In   the   second   case, 

4.  When  A  is  greater  than  p,  and  at  the  same 
less  than  both  B  and  180°  —  J?,  there  will  be  two  solu^ 
lions. 

5.  When  A  is  greater  than  p,  and  intermediate  in 
value  bettoeen  B  and  180°  —  B ;  or^  when  A  is  equal 
to     p,      there  will  be  but  one  solution. 

6.  When  A  is  greater  than  p,  and  at  the  same 
time  greater  than  both  B  and  1 80°  —  B ;  or^  wlien  A 
is  less   than     p,      there  will    be  no  solution. 

EXAMPLES. 

1.  Given  A  =  95°  16',  B  =  80°  42'  10",  and 
o   =    57°  38',       to  find      c,      by      and     C. 

Computing    p,      from  the   formula, 

log  sin^  =  log  sin  B  ■+■  log  sin  a  —  10  ; 

we  have,  p   =    56°  27'  52". 

The  smaller  value  of  p  is  taken,  because  a  is  less 
than   90°. 

Because  yi  >  p,  and  intermediate  between  80°  42'  10" 
and  99°  17'  50",  there  will,  from  the  fiflh  condition,  be  but 
a   single   solution. 

Applying   logarithms  to  Proportion  ( 1  ),  Art.  78,  we  have, 


(a.  c.)  log  sin  A  +  log  sin  B  +  log  sin  a  —  10  =  log  sin   b ; 

(a.  c.)  log  sin  A     (95°  IG')  0.001837 

log  sin  B     (80°  43'  10")  9.994257 

log  sin  a      (57°  38')  9.92GG71 

log  sin  b    ...    .  9.9227G5     .'.    b  =  6G°  49'  57". 


TRIGONOMETRY. 


99 


We  take  the  smaller  value  of  Z),  for  the  reason  that  A, 
being  greater  than  H,  requires  that  a  should  be  greater 
than     b. 

Appljnng  logarithms  to   Proportion  (12),   Art.  83,  we  have, 


(a.  c.)  log  cos  I  {A-ll)+\og  cos^  (^  +  i.')+log  tan  i  (a  +  Z/)— 10 

=  log  tan  I  c ; 
we  have, 

i  (.4  +B)  =  87°  59'  05",    ^  {a  +  b)  =  57°  13'  58", 
and,  i  {A  -  B)  =  7°  16'  55". 

(a.  c.)  log  cos  ^  (A-B)     .     (  7°  IC  55")     .  0.003517 

log  cos  -I  (A  +  B)     .     (87°  59'  05")     .  8.54G134 

log  tan  i  (a  +  h)     .     (57°  13'  58")     .  10.191352 

log  tan  Ic 8.740993 

.-.    i  c  =  3°  09'  09",     and  c  =  G°  18'  18". 

Applying  logarithms   to   the   proportion, 

sin   a    :     sin   c     :   :     sin   A     :     sin    C, 
we  have, 

(a.  c.)  log  sin  a  +  log  sin   c  +  log  sin  A  —  10  =  log  sin  (7; 
(a.  c.)  log  sin  a     (57°  38')       .    .    0.073329 


log  sin  c     (G°  18'  18")      .     9.040G85 

log  sin  A    (95°  IG')       .     .     9.9981  G3 

log  sin  C 9.112177 


C  =  7°  2G'  21". 


The   smaller   value   of     C     is    taken,   for  the    same    reason 
as  before. 

2.     Given     A   =  50°  12',     J5  =  58°  08',     and     a  =  62°42' 
to   find      b,     c,     and     C. 

79°  12'  10",  fll9°03'26",  f  130°  54' 28", 

b  =  -{  c  =  -l  C  = 


100°  47'  50", 


152°  14'  18", 


156°  15' 06". 


100  SPHERICAL 

CASE     m. 
Given  two  sides  and  their  included   angle. 

87.     The   reinaining   angles  are  found  by  n^ans  of  Yapier'a 
Analogies,   and   tlie   remaining   side,   .ts   in   the   precedu  g  cases. 

EXAMPLES. 

1.     Given         a    =    62°  38',         b    =     10^  13'  19",  and 

C    =    150°  24'  12",         to   find      c.      A,      and     J3. 

Aj^plying     logarithms     to     Proportions     (10)      and     (11), 
Art.  83,    we   have, 

^a.  c.)  log  cos  i  {a  +  h)  +  log  cos  4  (a  ~  h)  ■}-  log  cot  |  C  —  10 

=  log  tan  I  {A  +  B) ; 

(a.  c.)  log  sin  {a  +  I)  +  log  sin  I  (a  —  l)  +  log  cot  ^  C  —  10 

=  log  tan  ^  [A  -  B) ; 
we   have, 

^{a-l)  =  2G°  12'  20'V        ^  C  = '75°  12'  OG", 
and,  I  (a  +  L)  =  3G°  25'  39". 

va.  c.)  log  cos  I-  {a  +  b)      .     (36°  25'  39")     .  0.094415 

log  cos  i  {a  -  h)      .     (2C°  12'  20")     .  9.952897 

log  cot  ^  C      ...     (75°  12'  OG")     .  9.421901 

log  tan  I  {A  -[-  B) 9.4G9213 

.-.    ^  (A  +  B)  ^  1G°  24'  51' 

(a.  c.)  log  sin  h  (a  +  h)      .     (3G°  25'  39")     .  0.22G356 

log  sin  i  {a  -  h)     .     (2G°  12'  20")     .  9.G45022 

log  cot  ^  C      ...     (75°  12'  06")     .  9.421901 

log  tan  \  [A  -  B) 9.293279 

.-.    ^{A  -  B)  =  11°  06'  53". 


TRIGONOMETRY 


101 


The  grrater  angle  is  equal  to  the  half  sum  plus  the  half 
difference,  and  the  less  is  equal  to  the  half  sum  minus  the 
half  difference.       Hence,   we   have, 

A   =   27°  31'  44",         and         JJ  zz=   5°  17'  58". 


Applying   logarithms   to   the   Proportion   (13),    Art.  83,    wa 
lia^e, 

(a.  c.)  log  sin  ^{A-B)^  log  sin  \  (yl+^')  +log  <^an  \  {a-h)  -10 

=  log  tan  t|-  c, 

(a.  c.)  log  sin  ^  {A  -  B)  .     (11°  OC  53")     .  0.714052 

log  sin  -^  {A  +  B)  .     (1G°  24'  51")     .  9.451139 

log  tan  i  («  -    h)  .     (2G°  12'  20")     .  9.G92125 

log  tan  i  c 9.S58216 


.-.     -10  =  35°  48' 33",       and       c  =   71°  37' 06 


2,     Given        a    =    68°  46'  02",  b    =    37°  10',  and 

C  =  39°  23'  23",      to   find      c,      A,      and     i?. 

Ans.     ^  =  120' 59' 47",     i?  =  33°  45' 03",     c  =  43°  37' 38" 


3.     Given  a  =   84°  14'  29",         b  =  44°  13'  45",         and 

C  =   36°  45'  28",      to  find      A      and     B. 


Ans.     A   =   130°  05'  22",      ^  =  32°  26'  06 


CASE      IV. 


Given   two   angles   and  their  included  side. 
88.     The  solution  of  this   case  is  entirely  analogous  to  Case 

m. 

Applpng  logarithms   to   Proportions  (12)    and    (13),    Art 
83,   and  to   Proportion  (11),   Art.  83,   we   have, 


10?..-'  SPHERICAL 

(a.  c.)  log  cos  ^  {A  +  B)  +  log  cos  ^  {A  —  B)  -{-  log  tan  ^  c  —  10 

=  log  tan  -I-  {a  +  h)  ; 

(a.  c.)  log  sin  ^  {A  +  i>)  +  log  sin  -J-  {A  —  B)  +  log  tan  |  c  -  10 

=  log  tan  |-  [a  —  h) ; 

(a.  c.)  log  sin  {a  — I)  +  log  sin  {a  +  1)  +  log  tan  ^  (^  —  i>)  — 10 

=  log  cot  ^  C. 

The    application    of   tliese    formulas    are    sufficient    for    the 
Bolution   of  all   cases. 

EXAMPLES. 

1.  Given        A  =  81°  38'  20",        B  -   70°  09'  38",       and 
C  =  59°  16'  22",      to   find     (7,      a,      and     h. 

Ans.    C  =  64°  46'  24",     a  =  10°  04'  17'-,     b  =  63°  21'  27". 

2.  Given        ^   =  34°  15'  03",        i?  =  42°  15'  13",        and 
c  =   76°  35'  36",      to   find     C,      «,      and      b. 

Ans.    G  =  121°  36'  12",      a  ■=  40°  0'  10",      b  =  50°  10'  30". 


CASE      V. 

Given   the  three  sides,   to  find  the  remaining  parts. 

89.  The  angles  may  be  found  by  means  of  Formula  (3), 
Art.  81  ;  or,  one  angle  being  found  by  that  formula,  the  otlier 
two   may  be  found  by  means   of  Napier's  Analogies. 

EXAMPLES. 

1.  Given  a  =  74°  23',  b  =  35°  46'  14",  and  c  =  100°  30', 
to   find      A,      B,      and     C. 


TRIGONOMETRY. 


103 


Applying  logarithms  to  Formula  (3),   Art.  81,  wo  have, 

log  cos  ^A  =  10  +  -^[log  sin  ^s  +  log  sin  {^s  —  a) 

+  (a.  c.)  log  sin  J  +  (a.  c.)  log  sin  c  —  20]  ; 
or, 

og  cos  ^A  =  ^  [log  sin  ^s  +  log  sin  (|s  —  a) 

+  (a.  c.)  log  sin  b  +  (a.  c.)  log  sin  c], 
we    have, 

^5  =  105°  24'  07",  and  ^s  —  a  =  31°  01'  07  '. 

log  sin  ^s        •     •     •     (105°  24' 07")  •  9.984116 

log  sin  (is  -a)      •     (  31°  01'  07")  •  9.712074 

(a.  c.)  log  sin  6     ....     (  35°  46'  14")  •  0.233185 

(a.  c.)  log  sin  c (100°  39')  0.007546 


log  cos  ^A 


2)19.936921 
9.968460 


.-.     ^A  =  21°  34'  23",      and      A  =  43°  08'  46". 

Using  the  same  formula  as  before,  and  substituting  J3  for 
Ay  b  for  a,  and  a  for  5,  and  recollecting  that 
\8  —  b  =  09°  37'  63",      we   have, 


log  sin  ^s        .     ' 
log  sin  (^5  —  h) 
(a.  c.)  log  sin  a  .    •    • 
(a.  c.)  log  sin  c    •     •     • 


(105°  24'  07")  .  9.984116 
(  69°  37'  53")  .  9.971958 

.  (74°  23')  .  .  0.010336 

.    (100°  39')     .     •     0.007546 

2)19.979956 


log  cos  \B 


9.989978 


.-.     \B  =   12°  15' 43",      and      B  =  24°  31' 26' . 


Using  the  same  formula,  substituting  C  for  A^  c  for  a, 
and  a  for  c,  recollecting  that  \s  —  c  =  4°  45'  07",  we 
have, 


104         SPHERICAL     TRIGONOMETRY. 

log  sin  is       •     •      (105°  24'  07")  9.984116 

log  sin  (Is -c)      •     (4°  45' 07")     •  8.918250 

(a.  c.)  log  sin  a   .     •     •     •     (74°  23')     •     •     •     0.016336 

(a.  c.)  log  Bin  b    '     •     •        (35°  46'  14")     •     •     9.233185 

2)19.151887 
log  COS  |C 9.575943 

.-.     I C  =   67°  52' 25",       and       C  =   135°  44'  50" 

2.     Given      a  =  56°  40',      b  =  83°  13',     and      c  =  114°  30'. 
Am.    ^  =  48°  31'  18",    i?  =  62°  55' 44",    C  =  125°  18' 56". 

CASE      VI. 
The  tJiree  angles  being  given^   to  find  the   sides. 

90.     Tlie   solution   in  tliis  case  is  entirely  analogous  to  the 
preceding   one. 

Applying   logarithms  to   Formula  ( 2 ),   Art.  82,   "\ve    have, 

log  cos  la  —  i  [log  cos  {\S  —  i?)  +  log  cos  {^S  —  C) 

4-  (a.  c.)  log  sin  i?  +  (a.  c)  log  sin  C  \. 

In  the   same  manner   as  before,   we   change   the   letters,   to 
suit   each   case. 

EXAMPLES. 

1.  Given     A  =  48°  30',     ^  =  125°  20',     and     C  =  62°  54'. 
Ans.     «  =  5G°30'30",     Z*  =  114°  29' 58",     c    =  83°  12' 06" 

2.  Given       A   =   109°  55' 42",       J?  =   116°  88' 33",       and 
C  =   120°  43'  37",       to    find      a,       b,      and      c. 

A71S.     a  =  98°  21' 40",     J  =  109°  50' 22",     c  =  115°  13' 28". 


MENSURATION. 


91.  Mexsukation  is  that  branch  of  Mathematics  which 
treats   of  the   measurement   of  Geometrical   MasTnituJes. 

92.  The  measurement  of  a  quantity  is  the  operation  of 
finding  how  many  times  it  contains  another  quantity  of  the 
same  kind,  taken  as  a  standard.  This  standard  is  called  the 
unit   of  measure. 

93.  The  unit  of  measure  for  surfaces  is  a  square^  one 
of  whose  sides  is  the  linear  unit.  The  unit  of  measure  for 
volumes   is   a   ciihe^   one   of  whose   edges  is  the   linear   unit. 

If  the  linear  imit  is  one  foot,  the  superficial  unit  is  on^ 
square  foot,  and  the  unit  of  volume  is  one  cubic  foot.  If 
the  linear  unit  is  one  yard,  the  superficial  unit  is  one  square 
yard,   and   the   unit   of  volume   is   07ie  cubic  yard. 

94.  In  Mensuration,  the  term  product  of  two  lines,  is 
used  to  denote  the  product  obtained  by  multiplying  the 
number  of  linear  units  in  one  line  by  the  number  of  linear 
units  in  the  other.  The  term  product  of  three  lines,  is  used 
to  denote  the  continued  product  of  the  number  of  linear 
units   in    each    of  the    three   lines. 

Thus,  Avheu  we  say  that  the  area  of  a  parallelogram  is 
equal  to  the  product  of  its  base  and  altitude,  we  mean  that 
the  number  of  superficial  units  in  the  parallelogram  is  eijurd 
to  the  number  of  linear  units  in  the  base,  nuilti])lii'd  by  the 
number   of  linear   units   in   the   altitude.      In   like   manner,    the 


106  MENSURATION 

number  of  units  of  volume,  in  a  rectangular  parallelopipedon, 
is  equal  to  the  number  of  superficial  units  in  its  base  Dulti- 
plied  bj  the  number  of  linear  units  in  its  altitudes.  a-»^ 
so    on. 

MENSUEATION      OF      PLANE      FIGURES. 
To  find  the  area   of  a  ijarallelogram. 

95.  From      the     principle     demonstrated     m     Book    IV., 
Prop,  v.,   we   have   the   following 

RULE. 

3IuUiply  the    base  by   the    altitude ;    the  product    will   be 
the  area  required. 

EXAMPLES. 

1.  Find  the  area  of  a  parallelogram,  whose  base  is   12.25, 
and   whose   altitude   is    8,5.  Ans.     104.125. 

2.  What   is   the   area    of    a    square,   whose    side   is    204.3 
feet?  Ans.     41738.49  sq.  ft. 

3.  How    many    square    yards    are    there    in    a    rectangle 
whose  base  is    6G.3  feet,   and   altitude   33.3  feet  ? 

Ans.     245.31  sq.  yd. 

4.  What    is    the     area     of    a    rectangular     board,    whose 
k-ngth   is   12.^  feet,   and   breadth   9   inches?  9|  sq.  ft 

5.  What   is   the   number   of    square    yards    in    a   parallel© 
gram,   whose   base   is   37   feet,   and   altitude    5    feet   3    inches  ? 

Ans.     2I-iV. 

To  find  the  area  of  a  plane   triangle. 

96.  First  Case.    When   the   base   and    altitude    are   given. 


OF    SURFACES. 


107 


From   the   principle   demonstrated   in    Book  IV.,    Prop.  VI., 
we   may    write   the   following 

EULE. 

Multlphj   the  lase  hy  half  the  altitude  ;   the  inoduct  wiU 
f»e   the  area   required. 

EXAMPLES. 

1.  Find  the   area   of   a   triangle,   whose  base   is    625,    and 
altitude   520  feet.  Ans.     102500  sq.  ft. 

2.  Find   the  area    of    a    triangle,   in   square   yards,   whose 
base  is   40,   and   altitude   30   feet.  Ans.     66f. 

3.  Find   the   area   of    a   triangle,   in    square    yards,   whose 
base   is   49,   and  altitude   25^   feet.  A7is.     68.7361. 

Second    Case.     When    two   sides    and    their   included   angle 
are    ffiven. 

Let  ABC  represent  a  plane  tri- 
angle, in  which  the  side  AB  =  c, 
BC  =  a,  and  the  angle  B,  are 
given.  From  A  draw  AIJ  perpen- 
dicular to  BC  ;  this  will  be  the 
altitude  of  the  triangle.  From  For- 
mula   (  1  ),  Art.  37,    Plane   Trigonometry,  we   have, 

AB  =  c  sin  B. 

Denoting   the   area   of   the   triangle   by    Q,     and    applying  the 
rule   last   given,    we    have. 


Q  = 


ac  sin  B 


2 


or. 


2Q  —  ac  sin  B, 


sin  B 


Substituting   for    sin  B^     — ^      (Trig,,  Art.  30),  and   applying 

logarithms,    we   have, 

log  (2§)   =  log  a  +  log  c  -h  log  sin  i?  —  10  ; 


i08 


MENSURATION 


hence,   Ave   may   write   the  followiug 

KULE. 

Add  together  the  logarithnis  of  the  two  sides  and  the 
logca'iihmic  sine  of  their  included  angle  /  from  this  sum 
uihtracl  10  ;  tJie  remainder  loill  he  the  logaritlim  of  doiihlc 
the  area  of  the  triangle.  Find,  from  the  table,  the  nimiher 
answering  to  this  logarithm,  and  divide  it  by  2  ;  the  quotient 
will  be   the   required  area. 

EXAMPLES. 

1.  "What  is  the  area  of  a  triangle,  in  which  two  sides 
a  and  b,  are  respectively  equal  to  125.81,  and  57.65,  and 
whose  included   angle     C,    is    57°  25'? 

Ans.     2Q  =  G111.4,       and      Q  =  3055.7     Am 

2.  What  is  the  area  of  a  triangle,  whose  sides  are  30 
and   40,   and   their  included   angle    28°  57'  ?         A)is.     290.427. 

3.  What  is  the  number  of  square  yards  in  a  triangle,  of 
which  the  sides  are  25  feet  and  21.25  feet,  and  their  included 
angle   45°  ? 


Ans.     20.8694. 


LEMMA. 

To  find  half  an  angle,   lohen   the   tJiree  sides   of  a  jplane   trir 

angle  are  given. 

97.  Let  ABC  be  a  plane  tri- 
angle, the  angles  and  sides  being  de- 
noted  as   in   the  fi^-ure. 

We  have  (B.  IV.,  P.  XH.,  XHI.), 


a2   =   i2   _}.   c2   ::^ 


2c .  AD. (1.) 

When   the   angle    A     is   acute,    we   have    (Art.  37), 

AD  =  -^  cos  J. ;        when  obtuse,      AD'  =  b  cos  CAD'. 


OF     SURFACES. 


109 


But  as    CAD'    is  the  supplement,  of  the  obtuse   angle    A, 

cos  CAD'  =  —  cos  -4,        and        AD'  =  —  b   cos  A. 

Either   of  tliese  values,  being   substituted    for     AD^      in    ( 1 ), 
gives, 


whence, 


a^  =  J2  _|_  c2  _  2bo  cos  A  ; 

^2   +    c2    —   «2 


COS^    = 


2  be 


(2.) 


If    ■we    add     1     to    both    members,    and     recollect    that 
1  +  cos  J.  =  2  cos^l^    (Art.  CO),   Equation   (4),   we  have,  , 


2  cos'^^A  = 


Ihc  +  b"^  +  c^—ji? 
2bG 


_  (^  +  g)^  -  «^   _    {h  -^  c  +  a)   {b  -\-  c  —  a)  ^ 
~  2bo  ~  2bc  * 


or. 


cos^  ^A   = 


(b  +  c  +o)    {b  -\-  c  —  a) 
Abo 


(3.) 


If  we  put        Z»  +  c  +  a  =  5,        we  have, 
b  -\-  c  +  a  _   J 


is, 


l  ^  c  —  a        , 
and,  — =  is  —  a; 


2  "■''  '  2 

Substituting  in    (3),   and   extracting    the   square   root. 


cos 


i^  =  V 


!\s{U-  a) 


be 


.     (4.) 


the   plus   sign,    only,  being  used,  smce    ^A  <  90°  ;    hence, 

The  cosine  of  half  of  either  angle  of  a  plane  triangle^ 
ie  equal  to  the  square  root  of  half  the  sum  of  the  three 
eideSy  into  half  that  sum  minus  the  side  02:>posite  the  angle^ 
divided  by  the  rectangle  of  the  adjacent  sides. 


By   applying  logarithms,   we   have. 


log  cos  \A  = 


\  [log  Is  +  log  {{s  —  a)  4-  (a.  c.)  log  b  +  (a.  c.)  log  c].     •     (  ii.) 

24 


110 


MENSURATION 


If  we  subtract  both   members    of   Equation   ( 2 ),    from    1, 
and  recollect  that     1  —  cos  yl  =  2  siu^  ^A   (Art.  6C.),  wo  have, 


2  ein'  ^A  — 


2bc  —  1)^  —  e  +  o?- 
25c 


-  a''  -  (^  -  c)^    _    (ff  4-  5  -  c)    (g  —  &  +  c) 
~  26c  ""  26c 

Placuig,    as  before,      a  •\-  h  -\r  c  =  s,      we  have, 


(5. 


a  +  b  —  c          1                        ,           a  —  b  -{-  c         ,        , 
2 =    i«  -  «»  and,  ~ =  t«  -  J. 

Substituting  in   ( 5 ),   and  reducing,  we  have, 


hence. 


sin  ^A 


/{js-b)    {js^ 
~    V  be 


(6.) 


The  sine  of  half  an  angle  of  a  plane  triangle,  is  equal 
to  the  square  root  of  half  the  sicm  of  the  three  sides,  minus 
one  of  the  adjacent  sides,  into  the  half  sum  minus  the 
other  adjacent  side,  divided  by  the  rectangle  of  the  adjacent 
aides. 

Applying   logarithms,   we   have,    • 

log  Bm^A   =  ^  [log  {^s  —  J)  +  log  {y  -  c) 

+  (a.  c.)  log  b  +  (a.  c.)  log  c].     (SD.) 


Third  Case.    To   find  the   area    of    a    triangle,    when    tha 
hree   sides   are   given. 

Let  ABC  represent  a  triangle 
whose  sides  a,  b,  and  c  are  given. 
From  the  principle  demonstrated  in 
the  last  case,   we  have. 


Q  =  \bc  sin  A. 


OF     SURFACES.  Ill 

But,   from   Formula   (ii'),   Trig.,  Art.  GG,    wc   have, 

sin  -4  =  2  sin  ^A   cos  ^A  ; 

Nvhence, 

Q  —  be  sin  \A    cos  ^A. 

Substituting    for      sin  ^A     and      cos  \A^      their   values,    taken 
iVuiu   Lemma,    and   reducing,   we  have. 


hence,   we   may   wi'ite   the   follo^ving 

BULK. 

Fi7id  half  the  sum,  of  the  three  sides,  and  from  it  subtract 
each  side  separately.  Find  tJie  co7ititiued  jyoduct  of  the  half 
sum.  and  the  three  remainders,  and  extract  its  square  root  /  the 
result  will  be  the  area  required. 

It  is  generally  more  convenient  to  employ  logarithms  ;  for 
this  purpose,  applying  logarithms  to  the  last  equation,  we  have, 

log  Q  =  I  [log  U  +  log  {Is  -  a)  -I-  log  {Is  -  b)  +  log  {U-c)] 

hence,  we   have   the   following 

RULE. 

Find  the  half  sum  and  the  three  remainders  as  before,  then 
find  the  half  sum  of  their  logarithms  ;  the  number  correspond- 
ing to  the  resulting  logarithm  xnill  be  the  area  required, 

EXAMPLES. 

1.  Find  the  area  of  a  triangle,  whose  sides  are  20,  30, 
and   40. 

"We  have,  \s  =  45,  \s—a  =  25,  ^5  — 5  =  15,  |s  — c  =  5 
By  the   first   rule, 

Q    =    -v/45   X  25   X   15   X  5     =     290.'t737     Am. 


By   the   second  rule, 

log  is       ... 

.     (45) 

log  i^s-a)      ' 

.     (25) 

log  iks  -  h)      . 

•     (15) 

log  (is  -  c)      • 

•      (5) 

112  MENSURATION 

•     .     .  1.653213 

.     .     .  1.397940 

.     .     •  1.176091 

.     .     .  O.G08970 

2  )4. 9202 14 

log  (2 2.4G3107 

.'.      (2  =  290-4737     Ans. 

2.  How  many  square  yards  are  there  in  a  triangle,  whose 
ffldes   are   30,   40,   and   50  feet  ?  Ans.     66|. 

To  find  the   area   of  a   trapezoid. 

98.  From  the  princijDle  demonstrated  in  Book  r\^.,  Prop. 
VII.,    we   may  write  the   following 

RULE. 

Find  half  the  simi  of  the  lyaraUel  sides^  and  mxdtiply  it 
hy  the  altitude  ;    the  product  will  be  the   area  required. 

EXAMPLES. 

1.  In  a  trapezoid  the  parallel  sides  are  750  and  1225, 
and  the  perpendicular  distance  between  them  is  1540  ;  what 
is   the    area  ?  A7is.   1520750. 

2.  How  many  square  feet  are  contained  in  a  planlc,  whoso 
length  is  12  feet  6  inches,  the  breadth  at  the  greater  end  15 
inches,  and  at  tlie  less  end   11  inches?  Ans.     IS-J-J. 

3.  How  many  square  yards  are  there  in  a  trapezoid, 
whose  parallel  sides  are  240  feet,  320  feet,  and  altitude  GO 
feet  ?  Ans.     20531  sq.  yd 

To  find  the  area  of  any   quadrilateral. 

99.  From   what   precedes,   we   deduce   the   following 


OF     SURFACES. 


113 


RULE. 

Join  the  vertices  of  ttco  o2'>posite  anrjles  hy  a  diagonal ; 
from  each  of  the  other  vertices  let  fall  perjoeyullcidars  upon 
this  diagonal  /  vxxdtlphj  the  diagonal  hy  half  of  the  sum 
of  the  j^:)eJ2Je«c?/cw^a;'5,  and  the  2^^'oduct  will  be  the  area  re- 
quired. 

EXAMPLES. 


1.  What  is  the  area  of  the  quad- 
rilateral AB  CD^  the  diagonal  A  G 
being  42,  and  the  perpendiculars  Dg^ 
J3b,     equal   to    18   and    16  feet  ? 

A71S.     114  sq.  ft. 


2.  How  many  square  yards  of  paving  are  there  in  the 
quadrilateral,  "whose  diagonal  is  Go  feet,  and  the  two  perpen- 
diculars   let    fall    on   it    28    and    33.^   feet  ?  Ans.     222^,. 


To  find  the   area  of  any  polygon. 
100.     From   what   precedes,    -we   have   the   following 

11  U  L  E . 

Draw  diagoncds  dividing  the  proposed  polygon  into  tror 
pezoids  and  triangles  :  then  find  the  areas  of  these  figures 
separately^  and  add  them  together  for  the  area  of  the  whoU 
polygon. 

EXAMPLE. 

1.  Let  it  be  required  to  de- 
termine tlie  area  of  the  polygon 
ABCDE^     having   five   sides. 

Let  us  suppose  that  we  have  mea- 
sured  the   diagonals   and    perpendicu- 
lars,   and    found        AC  =  3G.21,       EC  =   39.11, 
Bd  =  1.2Q,     Aa  =  4.18  :     required  the  area, 


Bb  =  4 
A?is.  296.1292. 


114 


MENSURATION 


To  find  the  area  of  a  regular  polygon. 

101.  Let  AB^  denoted  by  s,  re- 
present one  side  of  a  regular  polygon, 
wbose  centre  is  C.  Draw  CA  and 
CJ5,  and  from  G  draw  CD  i^erpen- 
dicular  to  AB.  Then  will  CD  be  the 
apothem,  and  we  shall  have  AD  =  BD. 

Denote  the   number  of  sides   of  the  polygon   by     n  ;     tlien 

860° 
will    the    ano^le     ACB,      at    the    centre,    be    equal    to      , 

(B.  v.,  Page  138,  D.  2),   and  the   angle   A  CD,    which  is  half 

180° 
of  ACB,    will   be   equal  to     • 

In  the  right-angled  triangle  ADC,  we  shall  have,  For- 
mula  (3),   Art.  37,    Trig., 

CD  =  is  tan  CAD. 
But     CAD,    being    the     complement     of     A  CD,      we    have, 

tan  CAD  =  cot  A  CD  ; 

180° 
hence,  CD    =    ^s  cot  , 

a  formula  by  means  of  which  the  apothem  may  be  computed. 
But   the   area  is  equal  to  the   perimeter  multiplied  by  half 
the   apothem   (Book  V.,  Prop.  VIII.)  :    hence   the   following 

KULK 

JFind  the  apothem,  bg  the  j^recedlng  formula ;  multiply 
the  perimeter  bg  half  the  apothem  ;  the  product  will  he  tlie 
area  required. 

EXAMPLES. 

1.     What   is  the  area  of  a  regular  hexagon,  each   of  whose 
sides  is   20  ?       "We   have, 

Ci>  =  10  X  cot  30°  ;      or,     log  CD  =  log  10  +  log  cot  30° -10 

1.000000 


log  is 


log  cot- 


180° 


11 


(10) 
(30°) 


10.238561 


log  CD 1.238561 


CD  =   17.3205. 


OF     SURFACES, 


115 


The  perimeter  is  equal  to    120  :  hence,  denoting  the  area  by   Qy 

120  X  17.3205 


Q 


—    1039.23     Ans. 


2j    What   is   tlie   area   of  an   octagon,   one    of  whose   sides 


is   20? 


A71S.     1931.36886. 


The  areas  of  some  of  the  most  important  of  the  regular 
polygons  have  been  computed  by  the  preceding  method,  on 
tlie   sujijDosition  that   each   side   is  equal   to    1,    and  the  results 


are   given   in   the   following 


TABLE, 


NASIKS. 

SIDES. 

AKKAS. 

NAMES. 

SIDES. 

AREAS. 

Triangle,     . 

.     3     . 

.     0.4330127 

Octagon,     . 

.        & 

.     4.82S4271 

Square, 

.     4     . 

.     1.0000000 

Nonagon,    . 

.     9 

.     6.18185^12 

Pentagon,  . 

.     5     . 

.     1.7204V74 

Decagon,    . 

.  10 

.     .     7.6942088 

Hesagon     . 

.     6     . 

.     2.5980762 

Undecagon, 

.  11 

.     9.3656399 

Heptagon   . 

.     7     . 

.     3.6339124 

Dodecagon, 

.   12 

.   11.1961524 

The  areas  of  similar  2:)olygons  are  to  each  other  as  the 
squares   of  their  homologous   sides    (Booh  IV.,  Prop.  XXVII.). 

Denoting  the  area  of  a  regular  polygon  whose  side  is 
5>  ^y  (?5  ^^^  that  of  a  similar  polygon  whose  side  is 
1,    by    T,    the   tabular  area,  we  have, 

hence,   the   foUoAving  k  u  l  e  . 

Multiply  the  corresponding  tabular  area  by  the  square  of 
the  giveyi   side  /    the  product  will  be   the  area  required. 

EXAMPLES. 

1.  \Yhat  is  the  area  of  a  regular  hexagon,  each  of  whose 
sides   is   20  ? 

We  have,     T  =  2.598 07G2,       and     s^   _    400  ;    hence, 

Q   =   2.5980762   X  400   =   1039.23048     Ans. 


116  MENSURATION 

2.     Find   tlie   area  of  a  pentagon,   AvLose   side   is    2b. 

Ans.     1075.298375. 

S.     Find  the  ai-ea   of  a   decagon,    whoso   side   is    20. 

Ans.     3077.68352, 

To  Jincl  the  circumference  of  a  circle,  when  the  diaineter    is 

fjioen. 

102.  From   the  principle   demonstrated   in  Book  V.,  Prop. 
XVI.,   we   may  write   the   following 

BULK. 

Multiply   the  given   diameter   by    3.1416  ;    tJie  product  wiU 
be  the  circumference  required. 

E  X  A  ai  p  L  i:  s . 

1.  "What   is   the   circumference   of  a   circle,  whose  diameter 
is   25  ?  Ans.     78.54. 

2.  If   the   diameter    of   the    earth   is   7921    miles,   what    is 
the   circumference?  Ans.     24884.6136. 

To  find  the   diameter   of  a  circle,   ichen  the  circumference  is 

given. 

103.  From  the  preceding  case,  we  may  write  the  foUowuig 

RULE. 

Divide    the    given    circumference    hy    3.1416  ;    the   guotierU 
will  be   the   diameter  required. 

EXAJIPLES. 

1.  What   is   the  diameter  of  a   circle,  whose  circumference 
is    11G52.1944  ?  Ans.     3709. 

2.  What  is  the  diameter   of  a   circle,   whoso  circumferenca 
IB   6850  ?  Ans.     21S0.4J 


OF     SURFxiCES.  117 

To   fi)id    the    length    of  an    arc    containing  any  number   of 

degrees. 

104.  The  length  of  an  arc  of  1°,  in  a  circle  v\'l30se 
diameter  is  1,  is  equal  to  the  cireuinfereiice,  or  3.1416 
jirided  by  3G0  ;  that  is,  it  is  equal  1o  O.O0872GG  :  hence, 
the  length  of  an  arc  of  n  degrees,  Avill  he,  n  x  0.00872G6. 
To  find  the  lengtli  of  an  arc  containing  n  degrees,  when 
the  diameter  is  d^  we  employ  the  piincijile  demonsirated  in 
Book  v..  Prop.  XIII.,  C.  2 :   hence,  we  may  write  tlie  following 

n  u  L  E  . 
Midtiply   the    oiumher   of  degrees   hi   the   arc   hg    .OOS72CG, 
and  the    product    hg   the    diameter    of  the    circle  /    the  result 
will  be   the  length  retjuircd. 

EXAMPLES. 

1.  "What  is  the  length  of  an  arc  of  30  degrees,  the 
diameter   heing    18   feet?                                     Ans.  4.7123C4  ft. 

2.  What  is  the  length  of  an  arc  of  12°  10',  or  ]2a°,  the 
diameter    being    20   feet  ?                                     Ans.  2.123-472  ft. 


o 


To  find  the  area   of  a  circle. 

105.     From   the   principle   demonstrated   in   Book  V.,  Prop. 
XV.,   we   may   write  the   following 

K  U  L  E  . 

Multiphj   the  square   of   the    radius    by    3.1416  ;    the  pro- 
duct will  be  the  area  required. 

EXAMPLES. 

1.  Find   the   area   of  a   circle,  whose   diameter  is    10,   and 
circumference    31.416.  Ans.     78.54. 

2.  How   many   square   yards   in    a     circle   Avhose     diameter 
is   3^  feet?  Ans.     1.0G901G. 

3.  What   is   the  area    of    a    circle   whose    circumference   is 
12   feet?  Ans.     11. 4 5 95. 


118 


MENSURATION 


To  find  the   area  of  a   circular  sector. 

106,     From   the   principle   demonstrated   in   Book  V.,   Prop, 
XIV.,    C.  1  and  2,    we   may  write   the   follo^ving 

RULE. 

I.     Multiply  half  the  arc  hy   the  radius  ;    or, 
n.     Find  the   area   of  the  lohole  circle^   hy   the    last  rah  ; 
then  write  the  2^'i'oportio7i,  as   3G0   is  to  the  number  of  degrees 
in   the  sector^   so  is  the  area  of  the  circle   to   the  area  of  the 
sector. 

EXAMPLES. 

1.  Find  the   area   of  a   circular   sector,  whose   arc   contains 
18°,  the  diameter  of  the  circle  being  3  feet.  0. 353-43  sq.  ft. 

2.  Find   the    area   of   a   sector,    whose   arc   is   20  feet,    the 
radius   beino;   10.  Ans.     100. 

3.  Required   the   area   of  a   sector,   Avhose   arc_  is   147°  29', 
and   radius   25  feet.  Ans.     804.3986  sq.  ft. 


To  find  the  area   of  a  circular  segment. 

107.  Let  AB  represent  the  chord 
corresponding  to  tlie  two  segments 
ACB  and  AFB.  Draw  AE  and 
BF.  The  segment  ACB  is  equal  to 
the  sector  FA  CB,  minus  the  triangle 
AFB.  The  segment  AFB  is  equal 
lo  the  sector  FAFB,  plus  the  tri- 
angle AFB.  Hence,  we  have  the  fol- 
lowing 

RULE. 

Find  the  area  of  the  corresponding  sector^  and  also  of 
the  triangle  formed  hy  the  chord  of  the  segment  and  the 
two  extreme  radii  of  the  sector ;  suhtract  the  latter  from  the 
former  when  the  segment  is  less  than  a  semicircle^  and  take 
their  sum  xohen  the  segment  is  greater  than  a  semicircle ; 
the  result  will  he   the  area  required. 


OF     SURFACES.  119 

EXAMPLES. 

1.  Find  the  area  of  a  segment,  Aviiose  chord  is  12  and 
the   radius    10. 

Solving  the  triangle  AEB,  Ave  find  the  angle  AEli  is 
equal  to  73°  44',  the  area  of  the  sector  JEA  CB  equal  to 
34.35  and  the  area  of  the  triangle  AEB  equal  to  48  ; 
lence,    the   segment    ACB     is   equal   to    10.35     Ans. 

2.  Find  the  area  of  a  segment,  whose  height  is  18,  the 
diameter  of  the    circle   heing   50.  Ans.     636.4834. 

3.  Required  the  area  of  a  segment,  whose  chord  is  16, 
the    diameter  being   20.  Ans.     44.764. 

To  find  the  area    of   a    circular    ring  contained  hetween    the 
circumferences   of  two   concentric  circles. 

108.  Let  B,  and  r  denote  the  radii  of  tlie  two  circles, 
R  behig  greater  than  r.  The  area  of  the  outer  circle  is 
R-  X  3.141G,  and  that  of  the  inner  circle  is  r"^  X  3.1416  ; 
hence,  tlie  area  of  tlie  ring  is  equal  to  {R"^  —  r"^)  X  3.1416. 
Hence,   the   following 

K  U  L  E  . 

Find  the  difference  of  tJie  squares  of  the  radii  of  the 
two  circles^  and  mxdtiphj  it  hy  3.1416  ;  tlie  product  will  be 
the   area  required. 

EXAMPLES. 

1.  The  diameters  of  two  concentric  circles  being  10  and 
6,  required  the  area  of  the  ring  contained  between  their 
circumferences.  Ans.     50.2656. 

2.  What  is  the  area  of  the  rhig,  when  the  diameters  of 
th3    circles   are    10   and   20  ?  A7is.     235.62 


120  MENSURATION 

MENSDP.ATIO:?^      OF      BEOKEN      AND      CUKVED      SUEFACES. 
To  find  the  area  of  the  entire  surface  of  a  right  prism. 

109.  From  the  principle  demonstrated  in  Book  VII.,  Prop, 
I.,    Avo   may   "svrite   the   following 

K  u  L  E  . 

Multiply  the  perimeter  of  the  base  by  the  altitude^  the  pro- 
duct 10 ill  be  the  area  of  the  convex  surface  y  to  this  add  the 
areas  of  the  two  bases  /    the  residt  xoill  be   the  area  required. 

EXAMPLES. 

1.  Find  the  surface  of  a  cube,  the  length  of  each  side 
being   20    feet.  Ans.     2400  sq.  ft. 

2.  Fmd  the  whole  surface  of  a  triangular  prism,  whose 
base  is  an  equilateral  triangle,  having  each  of  its  sides  equal 
to    18    inches,    and   altitude    20    feet.  Ans.     91.949  sq.  ft. 

To  find  the   area  of  the  entire  surface   of  a  right  ivjramid. 

110.  From  the  principle  demonstrated  in  Book  VII.,  Prop. 
IV.,  we   may  write   the   following 

EULE. 

Mitltiply  the  perimeter  of  the  base  by  half  the  slant 
height;  the  product  will  be  the  area  of  the  convex  surface; 
to  this  add  the  area  of  the  base;  the  result  icill  be  the  area 
rcqidrcd. 

EXAMPLES. 

1.  Find  the  convex  surface  of  a  right  triangular  pyramid^ 
the  slant  height  being  20  feet,  and  each  side  of  the  base 
3    feet.  Ans.     90  sq.  ft 

2.  Wliat  is  the  entire  surface  of  a  right  pyramid,  whose 
slant  height  is  15  feet,  and  the  base  a  pentagon,  of  which 
each   side   is   25   feet  ?  Ans.     2012.798  sq.  ft. 


OF    SURFACES.  121 

To    find    the    area     of    the    convex    surface    of    a    fnistum    of    a 

ric/ht  ^)y?'a??i2rf. 

nil  From  the  principle  demonstrated  in  Boole  XII.,  Prop. 
JV.,  C,  we    may    Avi-itc   the   following 

RULE. 

Mil  It!])'//  the  half  sum  of  the  perimeters  of  the  two  bases 
by    the   slant   height ;    the  j^^'oduct   tuill   be    the   area   reqxdrcd. 

EXAMPLES. 

1.  How  many  square  feet  arc  there  in  the  convex  surface 
of  the  frustum  of  a  square  pyramid,  whose  slant  height  is  10 
feet,  each  side  of  the  lower  base  3  feet  4  inches,  and  each  side 
of   the    upper   base    2    feet    2    inches?  Ans.     110  sq.  ft. 

2.  "\That    is    the    convex    surface    of    the    frustum    of    a    hep- 
tagonal    pyramid,    whose    slant    height    is    55    feet,    each    side    of 
the    lower    base    8    feet,    and    each    side    of  .  the    upper    base    4 
feet?  Ans.     2310  sq.  ft. 

112i  Since  a  cylinder  may  be  regarded  as  a  prism  whose 
base  has  an  infinite  number  of  sides,  and  a  cone  as  a  pyra- 
mid whose  base  has  an  infinite  number  of  sides,  the  rnlcs  just 
given,  may  be  applied  to  find  the  areas  of  the  surfaces  of  right 
cylinders,  cones,  and  frustums  of  cones,  by  simply  changing  the 
terra  perimeter,    to    circumference. 

EXAMPLES. 

1.  ^Yhat  is  the  convex  surface  of  a  cylinder,  the  diameter 
of   whose    base    is    20,  and    whose    altitude    50?  Ans.     3141.0. 

2.  "What  is  the  entire  surface  of  a  cylinder,  the  altitude 
being   20,    and  diameter   of   the    base    2    feet?  131.9472  sq.  ft. 

3.  Eequired  the  convex  surface  of  a  cone,  whose  slant 
height   is    50    feet,    and    the    diameter    of    its   base    Si    feet. 

Ans.     CG7.59  sq.  ft. 


122  MENSURATION 

4.  Required  the  entire  surflico  of  a  cone,  whose  slant 
he'vAit  is    3G,   and  the   diameter   of   its  base   18   feet. 

Ans.     1272.348  bq.  ft. 

5.  Find  the  convex  surface  of  the  frustum  of  a  cone,  the 
fclant  height  of  the  frustum  being  12^  feet,  and  the  circum- 
ferences  of  the   bases   8.4   feet   and   6    feet.         Ans.     90  sq.  ft. 

6.  Find  the  entire  surface  of  the  frustum  of  a  cone,  the 
slant  height  being  16  feet,  and  the  radii  of  the  bases  3  feet, 
and   2    feet.  ^ns.     292.1688  sq.  ft. 

To  find  the  area   of  the  surface  of  a  sphere. 

113.  From  the  principle  demonstrated  in  Book  VIII , 
Prop.  X.,  C.  1,    we   may   write   the   following 

RULE. 

Find  the  area  of  one  of  its  great  circles,  and  midtq^ly 
It  hj  4:  ;    the  product  will  be   the  area  required. 

EXAMPLES. 

1.  What  is  the  area  of  the  surface  of  a  sphere,  whose 
radius   is   16  ?  ^ns.     3216.9984. 

2.  What  is  the  area  of  the  surftice  of  a  sphere,  whose 
radius  is   27.25  -^ns.     9331.3374. 

To  find  the   area  of  a  zone. 

114.  From  the  principle  demonstrated  in  Book  VIII.. 
I\op.  X.,  C.  2,   we   may  Avrite   the   following 

K  U  L  E. 

Find  the  circumference  of  a  great  circle  of  the  sphere, 
and  midtlply  it  by  the  altitude  of  the  zo7ie ;  the  2^'>'oduct 
will  be   the  area  required. 


OF     SURFACES.  123 

EXAMPLES. 

1.  The  diameter  of  a  sphere  being  42  inches,  what  is 
the   area   of  the  surface  of  a  zone  whose  altitude   is   9   inches. 

A71S.     1187.5248  sq.  in. 

S".  If  the  diameter  of  a  sphere  is  12^^  feet,  what  will  he 
the  suvflice  of  a  zone  whose  altitude  is  2  feet  ?         78.54  sq.  ft» 

To  find  the  area  of  a  spherical  'polygon. 

115.  From  the  principle  demonstrated  in  Book  IX.,  Prop. 
XIX.,  we  may  write   the  followmg 

KU  L  E. 

From  the  sum  of  the  angles  of  the  polygon^  subtract  180° 
taken  as  many  times  as  the  polygon  has  sides^  less  two^ 
and  divide  th6  remainder  hy  90°  /  the  quotient  will  he  the 
spherical  excess.  Find  the  area  of  a  great  circle  of  the 
sphere,  and  divide  it  hy  2  y  the  quotient  xcill  he  the  area 
of  a  tri-rectangular  triangle.  Multiply  the  area  of  the  tri- 
rectangular  triangle  hy  the  spherical  excess,  and  the  product 
wiU  he   the  area  required. 

This  rule  applies  to  the  spherical  triangle,  as  well  as  to 
any   other   spherical   polygon. 

EXAMPLES. 

1.  Required  the  area  of  a  triangle  described  on  a  sphere, 
whose  diameter  is  30  feet,  the  angles  being  140°,  92°,  and 
60°.  Ans.     4Y1.24  sq.  ft 

2.  "What  is  the  area  of  a  polygon  of  seven  sides,  do 
scribed  on  a  sphere  whose  diameter  is  17  feet,  the  sum  of 
Ihc   angles   heme:   1080°  ?  Ans.     226.98 

3.  "Wliat  is  the  area  of  a  regular  polygon  of  eight  sides, 
described  on  a  sphere  whose  diameter  is  30  yards,  each  an- 
gle  of  the   polygon  being   140°  ?  Ans.     157.08  sq.  yds. 


124  MENSURATION 

MENSURATIO]Sr      OP     VOLUMES. 

To  f.nd  the  volume  of  a  2}rism. 

116.     From    the    principle     demonstrated    in     Book    VII., 
Prop.  XIV.,   we   may   Avrite   the   following 

EULE. 

Multijoly  the  area  of  the  base  hy  the  altitude  ;  the  pro- 
duct  will  he   the  volume  required. 

EXAMPLES. 

1.  What  is  the  volume  of  a  cube,  whose  side  is  24  inches? 

Ans.     13824  cu.  in. 

2.  How  many  cuhic  feet  in  a  block  of  marble,  of  -o-hich 
the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and 
height   or  thickness   2   feet   G   inches  ?  Ans.     21^  cu.  ft. 

3.  Required  the  volume  of  a  triangular  prism,  whose 
height  is  10  feet,  and  the  three  sides  of  its  triangular  base 
8,    4,    and   5   feet.  Ans.     60. 

To  find  the  volume  of  a  j^y^'cc^nid. 

.  117.     From  the  princij^le  demonstrated  in  Book  VII.,  Prop. 
XVn.,   Ave  may   write   the   following 

K  u  L  E  . 

Micltijjli/  the  area  of  the  base  by  one-third  of  the  alti- 
tude •    the  product  will  be  the  volume  required. 

EXAMPLES. 

1.  Required  the  volume  of  a  square  pyi-amid,  each  side 
of  its  base   being   30,   and   the   altitude   25.  Ans.     T500. 

2.  Find  the  volume  of  a  triangular  pyramid,  whose  alti- 
tude is  30,  and  each  side  of  the  base  3  feet.       38.9711  cu.  ft. 


OF     VOLUMES.  125 

3.  What  is  the  volume  of  a  pentagonal  pyramid,  its  alti- 
tude  being -12  feet,   and   each   side   of  its  base    2  feet. 

A71S.     27.5270  cu.  ft. 

4,  AVliat  is  the  volume  of  an  hexagonal  pyramid,  whose 
altitude  is   6.4  feet,  and   each   side   of  its  base   6  inches  ^ 

Ans.     1.38504  cu.  ft 

To  find  the  volume   of  a  frustum   of  a  j'^yi^cimid. 

118.     From  the  principle  demonstrated  iu  Book  VII.,  Prop., 
XVin.,  C,  we  may  write   the  following 

EULE. 

J^ind  the  sum,  of  the  iipper  base,  the  loicer  base,  and  a 
mean  proj)ortional  betxoeen  them  ;  mxdtlply  the  result  by  one- 
third  of  the  altitude  /   the  product  will  be  the  volume  required. 

EXAMPLES. 

1.  Find  the  number  of  cubic  feet  in  a  piece  of  timber, 
whose  bases  are  squares,  each  side  of  the  lower  base  being 
15  inches,  and  each  side  of  the  ujjper  base  6  inches,  the 
altitxade   being  24  feet.  Ans.     19.5. 

2.  Required  the  volume  of  a  pentagonal  frustum,  whose 
altitude  is  5  feet,  each  side  of  the  lower  base  18  inches,  and 
each  side  of  the  upper  base  G  inches.  A^is.     9.31925  cu.  ft. 

119.  Since  cylinders  and  cones  are  limiting  cases  of  prisms 
and  pyramids,  the  three  preceding  rules  are  equally  applicable 
to   them. 

EXAMPLES. 

1.  Required  the  volume  of  a  cylinder  whose  altitude  is 
12    feet,    and   the    diameter   of    its   base    15    feet. 

Ans.     2120.58  cu.  ft. 

2.  Required  the  volume  of  a  cylinder  whose  altitude  is 
20  feet,  and  the  circumference  of  Avhose  base  is  5  feet 
C   mches.  Ans.     48.144  cu.  ft. 

2& 


«26  MENSTTKATION 

3.  Eequirod  the  volume  of  a  cone  whose  altitaJe  is 
27     feet,    and    tlie    diameter    of   the    hase    10    feet. 

Ans.     706.86  cu.  ft. 

4.  Tlc]  aired  the  vohime  of  a  cone  whose  altitude  is 
,10|    feet,    and   the    circumference   of  its   base    9    feet. 

Ans.     22.5 G  en.  ft. 

5.  Find  the  volume  of  the  frustum  of  a  cone,  the  altitude 
ucmg  18,  the  diameter  of  the  lower  base  8,  and  that  of  the 
upper   base   4.  Ajis.     527.7888. 

6.  What  is  the  volume  of  the  frustum  of  a  cone,  the 
altitude  being  25,  the  circumference  of  the  lower  base  20, 
and  that   of  the   npper  base    10  ?  Ans.     464.216. 

7.  If  a  cask,  which  is  composed  of  two  equal  conic  frus- 
tnms  joined  together  at  their  larger  bases,  have  its  bung  dia- 
meter 28  inches,  the  head  diameter  20  inches,  and  the  length 
40  niches,  how  many  gallons  of  wine  will  it  contain,  there 
being    231   cubic   inches   in   a   gallon?  Ans.     79.0G13. 

To  find  the  volume  of  a  sphere. 

120.  From  the  principle  demonstrated  in  Book  Vill., 
Prop.  XIV.,   we   may  write   the  following 

RULE. 

Ciihe  the  diameter  of  the  sphere,  and  multqyJi/  the  result 
J)y  lY,  that  is,  hy  0.5230  ;  the  product  will  he  the  vohmie 
required. 

EXAMPLES. 

1.  What  is  the  volume  of  a  sphere,  whose  diameter  ia 
12  'i  Ans.     904.7808 

2.  What   is  the   volume   of   the   earth,   if  the   mean   diam 

eter  be   taken   equal  to   7918.7  miles. 

Ans.     259992792083   en.  miles 


OF    VOLUMES.  127 

To  find  the   voluuie   of  a  icedge. 

121.  A  Wedge  is  a  volume  bouml- 
ed  by  a  rectangle  ABCD^  called  the 
hack^  two  ti-apezoicis  ABHG^  JDCIIG^ 
called  faces,  and  two  triangles  ADG, 
CBH  called  ends.  The  Hne  GH,  in 
which  the  flices  meet,  is  called  the  edge. 
Tlie  two  faces  are  equally  inclined  to 
the  back,  and  so  also  are  the  two  ends. 

There  are  three  cases:  1st,  When  the  length  of  the  edge  is 
equal  to  the  length  of  the  back;  2d,  When  it  is  less;  and  3d, 
When  it  is  greater. 

In  the  first  case,  the  Avedge  is  a  right  prism,  whose  base  is 
the  triangle  ABG,  and  altitude  GH  or  AB :  hence,  its  volume 
is  equal  to  ADG  multiplied  by  AB. 

In  the  second  case,  through  II, 
the  middle  point  of  the  edge,  pass 
a  plane  IICB  perpendicular  to  the 
back  and  intersecting  it  in  the  line 
BC  parallel  to  AD.  This  plane 
will  divide  the  wedge  into  two 
parts,  one  of  which  is  represented 
by  the  figure. 

Through  G,  draw  the  plane  GNM  parallel  to  HCB,  and  it 
will  divide  the  part  of  the  Avedge  represented  by  the  figure  into 
the  right  triangular  prism  GISTM  —  J5,  and  the  quadrangular  pyr 
araid  ABJSfM—  G.  Draw  GP  perpendicular  to  NM:  it  will 
also  be  perpendicular  to  the  back  of  the  wedge  (B.  VI.,  P. 
.XVII.),  and  hence,  will  be  equal  to  the  altitude  of  the  wedge. 

Denote  AB  by  Z,  the  breadth  AD  by  b,  the  edge  GH  by 
?,  the  altitude  by  A,  and  the  volume  by  F";     then, 

AMz^L-l,   MB  =  Gn=  I,   and   area   NG3L  =  \hh  :    then 

Prism  =  \hhl\    Pyramid  =  b{L  -  l)^h  =  \bh{L  -  I),    and 
V=  \bhl  +  lhh{L  -l)  =  \bld  +  \bhL  -  \b]d  =  lb?i{l+2L). 

We  can  find  a  similar  expression  for  the  remaining  part  of  the 
wedge,  and  by  adding,  the  factor  within  the  parenthesis  becomen 
the  entire  length  of  the  edge  plus  twice  the  length  of  the  back. 


128 


MENSURATION 


In  the  tliird  case,  I  is  greater  than  Z,  and  denotes  the 
altitude  of  the  piisia  ;  tiie  vohinie  of  each  jiart  is  equal  to 
the  diflerence  of  the  prism  and  pyramid,  and  is  of  the  same 
form  as  before.      Hence,  the  following 

Wm.^.—Acld  twice  the  length  of  tlie  hack  to  the  length  of 
the  edge;  multijyly  the  sum  by  tJie  breadth  of  the  hack,  and 
that  restdt  by  one-sixth  of  the  altitude;  the  final  product  will 
he  the  volume  required. 

E  X  A  JI  I'  L  K  s . 

1.  If  the  back  of  a  wedge  is  40  by  20  feet,  the  edge 
35    feet,    and   the   altitude    10   feet,    what   is   the   volume? 

Ans.    3833.33  cu.ft. 

2.  What  is  the  volume  of  a  wedge,  whose  back  is  18  feet 
by  9,  edge  20  feet,  and  altitude  6  feet?  504  cu.ft. 

To  find  the  volume  of  a  j^rismoid. 

!22.     A  PrasMOiD  is    a  frustum   of  a  wedge. 

Let  Z  and  J]  denote  the 
length  and  breadth  of  the  lower 
base,  I  and  h  the  length  and 
breadth  of  the  npper  base,  31  and 
m  the  length  and  breadth  of  the 
section  equidistant  from  the  bases, 
a-nd   h    the  altitude  of  the  prismoid. 

Through  the  edges  X  and  /', 
let   a   plane   be    passed,   and    it   will 

divide   the    prismoid    into    two   wedges,   having   for  bases,    the 
bases   of  the   prismoid,  and   for   edges   the   lines    X     and    /'. 

The  volume  of  the  prismoid,  denoted  by  T^,  will  be 
equal  to  the   sum  of  the  volumes  of  the  two   wedges  ;    hence, 


r 


1\T 


or. 


V  =    -}Bh{l  +  2Z)  +  ibh{L  +  20  ; 

V  =   \h[2BL  +  2hl  +  BI  +  hL)  ; 


OF     VOLUMES.  129 

which  may  be   ■written   luulor  the   form, 

V  =   ih  [{BL  +  hl  +  Bl  +  bZ)  +  BZ  +  bl].  (ii.) 

Because   the   auxiliary   section    is    midway   between    the    b.u«ea, 
v\e  have, 


hence, 


231  z=  Z  +  I, 


and 


:m 


B  +  h; 


4Mm  :=   (L  +  l)  {J]  H-  ,';)    z=z  BL  ■{-  Bl  +  hZ  -f  U. 

Substituting  in    {■■^),   we   have, 

V  =   ih{BZ  +  U  +  AlLu). 

But  BZ  is  the  area  of  the  lower  base,  or  lower  section, 
U  is  the  area  of  the  upper  base,  or  upper  section,  and  3Im 
is   the   area   of  the  middle  section  :    hence,  the  followino- 

RULE. 

To  find  the  volume  of  a  lorlsmoid^  find  the  sum  of  the 
areas  of  the  extreme  sections  and  fortr  times  the  middle  sec- 
tion ;  multiply  the  residt  by  one-sixth  of  the  distance  between 
the   extreme  sections  ;    the  residt  will  be   the  volume  required. 

This  rule  is  used  in  computing  volumes  of  earth-work  in 
railroad  cutting  and  emhankment,  and  is  of  very  extensive 
application.  It  may  be  shown  that  the  same  rule  iiolds  for 
every  one  of  the  volumes  heretofoi'e  discussed  in  this  work. 
Thus,  in  a  i>yramid,  we  may  regard  the  base  as  one  extreme 
section,  and  the  vertex  (whose  area  is  0),  as  the  other 
extreme  ;  their  sum  is  equal  to  the  area  of  the  base.  The 
area  of  a  section  midway  between  between  them  is  equal  to 
one-fourth  of  the  base  :  hence,  four  times  the  middle  section 
is  equal  to  the  base.  Multiplying  the  sum  of  these  by  one- 
sixth  of  the  altitude,  gives  the  same  result  as  that  already 
found.  The  application  of  the  rule  to  the  case  of  cylinders, 
frustums  of  cones,  spheres,  &c.,  is  left  as  an  exercise  for  the 
student. 


130  MENSURATION 


EXA3IPLES. 


1.  One   of  tlie   bases  of  a  rectangular  prismoid  is    25    feet 
l»v    20,   the   other    15    feet   by    10,    and   the    altitude    12    feet 
required   the   volume.  Ans.     3700  cu.  ft. 

2.  What  is  the  volume  of  a  stick  of  hewn  tinihei-, 
whose  ends  are  30  inches  by  27,  and  24  mches  by  IS,  its 
leuirth   bein;];    24   feet  ?  Ajis.     102  cu.  11. 


ZilEXSUllATION      OF      REGULAR      POLYEDROIS'S. 

123.  A  llEGULxiK  PoLYEDKOX  is  a  polycdrou  bounded,  by 
equal  regular  polygons. 

The  polyedral  angles  of  any  regular  polyedron  are  all 
equal. 

124.  There  are  five  regular  i^olyedrons  (Book  YIL, 
Page   208). 

To  find    the    diedral    angle    betxceen   the   faces   of  a  recjidar 

polyedron. 

125.  Let  the  vertex  of  any  polyedral  angle  be  taken  as 
the  centre  of  a  sphere  whose  radius  is  1  :  then  will  this 
sphere,  by  its  intersections  with  the  faces  of  the  polyedral 
angle,  determine  a  regular  spherical  polygon  whose  sides  will 
be  equal  to  the  plane  angles  that  bound  the  polyedral  angle, 
and  whose  angles  are  equal  to  the  diedral  angles  between 
the  faces. 

It  only  remains  to  deduce  a  formula  for  finding  one 
angle  of  a  regular  si^hei-ical  polygon,  when  the  sides  are 
given. 


OF     rOLYEDRONS. 


131 


Let    ABCDjB    represent   a   regular   spherical    polygon,  and 
let    F   be  tlie  pole  of  a  small  circle  passing  through  its  verti- 
ces.      Suppose     P     to    be    connected 
with  each  of  the   vertices   by   arcs   of 
great    circles  :     there     will     tlius     be 
formed    as    many    equal    isosceles    tri- 
angles   as   the  polygon   has   sides,  the 
vertical    angle    in     each    being    equal 
to    360°    divided    by   the    number    of 
sides.      Through    F    draw    FQ     per- 
pendicular   to     AB :    then    will     A  Q 

be  equal  to   FQ.      If  we   denote   the   number   of  sides   by    n, 

360°  180° 

the   angle    AF  Q    will    be    equal   to 


or 


2n     '  n 

In  the  right-angled  spherical  triangle  AFQ,  "we  know  the 
base  AQ^  and  the  vertical  angle  AFQ\  hence,  by  Napier's 
rules  ^ov   circular  parts,   we   have, 

sin  (90°  —  AFQ)  =  cos  (90°  —  FA  Q)  co^  AQ\ 


or,   by  reduction,  denoting  the   side    AB    by    s,     and  the  an- 
gle    FAB,    byi^, 

180° 


cos 


whence. 


n 


=  sin  ^A  cos  \s  ; 


cos 


180° 


sin  ^A 


n 


cos  \% 


EXAMPLES. 


In   the   Tctraedron, 
180" 


n 


—  60°,      and       \s  =  30= 


In    the   Ilexaedron, 
180= 


11 


A  =   V0°  31'  42''. 


=  60°,      and       ^s  =  45°     .'      A  =  00°. 


132  MENSURATION 

lu   the   Octaedron, 
180= 


n 
In   tlie   Dodecaedron, 


45%     and      ^5  —     30^     ,'.    A   =  109^  28'  18" 


lOAO  • 

=  G0°,     and      U  =     54o     .-.    A   =  116°  33'  5 1".  f 


71 

In  the   Icosaedron, 
180° 


n 


36°,     and       ^5  =     30°     .-.    A   =   138°  1  l' 23". 


To  find  the  volume   of  a  regular  polyedron. 

126.  If  planes  be  passed  tlu'ough  the  centre  of  the  poly- 
edron and  each  of  the  edges,  they  will  divide  the  jiolyedron 
into  as  many  equal  right  pyramids  as  the  polyedron  has  faces. 
The  common  vertex  of  these  pyramids  avUI  be  at  the  centre 
of  the  polyedron,  their  bases  "will  be  the  faces  of  the  poly- 
edron, and  their  lateral  faces  Avill  bisect  the  diedral  angles 
of  the  polyedron.  The  volume  of  each  pyramid  will  be  equal 
to  its  base  into  one-third  of  its  altitude,  and  this  multiplied 
by  the  number  of  faces,  will  be  the  volume   of  the  polyedron. 

It  onlv  remains  to  deduce  a  formula  for  findinir  the  dis- 
tance   from  the   centre   to    one   face   of  the   polyedron. 

Conceive  a  perpendicular  to  be  drawn  from  the  centre  of 
the  polyedron  to  one  face  ;  the  foot  of  this  perpendicular 
will  be  the  centre  of  the  face.  From  the  foot  of  this  per- 
pendicular, draw  a  perpendicular  to  either  side  of  the  face 
in  which  it  lies,  and  connect  the  point  thus  determined  with 
the  centre  of  the  polyedron.  There  will  thus  be  formed  a 
right-angled  triangle,  whose  base  is  the  apothem  of  the  fixce, 
whose  angle  at  the  base  is  half  the  diedral  angle  of  the 
polyedron,  and  whose  altitude  is  the  required  altitude  of  the 
pyramid,  or  in  other  words,  the  radius  of  the  inscribed 
sphere. 


■# 


OF     rOLYEDRONS. 


133 


Denoting   the    i^crpendicular   Ly     P,     tlie   base   by     A,     and 

the    dieclrul    angle    by     A,     we   have  Formuhi    ( 3 ),    Art.  37, 

Trig., 

P  =  b  tan  ^A  ; 


out  b  is  the  apothem  of  one  face  ;  if,  therefore,  Ave  denote 
the  number  of  sides  in  tliat  foce  by  «,  and  the  length  of 
each   side   by    5,    vre   shall   have    (Art.  101,  Mens.), 

180° 


hs   cot 


71 


whence,   by   substitution, 


180° 
I*  =  is  cot  tan  iA  ; 


hence,  the  volume  may  be  computed.  The  volumes  of  all 
the  regular  polyedrons  have  been  computed  on  the  supposi- 
tion that  their  edges  are  each  equal  to  1,  and  the  results 
are   given   in   the   following 


TABLE. 

NAMES.  NO.    OP    FACES. 

Tetraedron, 4 

Hexaedron, 6 

Octaedron, 8 

Dodecaedron,      ....  12 

Icosaedron, 20 


TOI.UMES. 

0.1178513 
l.OOnoOOO 
0.4714045 

7.00;;  1  ISO 

2.181C950 


From    tlie    principles  demon.strated   in   Book  VII.,  we    ma) 
write    the   following 


KULE. 


To  find  th(^  volume  of  any  regular  pohjedro7i^  muUi2jbj 
the  cube  of  its  edge  by  the  corresponding  tabular  volume ; 
tJie  product   will   be   the   volume    required. 


134  MENSURATION. 

EXAMPLES. 

1.  What  is  the  volume  of  a  tetraedron,  whose  edge  is  15  ? 

Ans.     397.75. 

2.  What  is  the  volume  of  a  hexaedron,  whose  edge  is  12? 

Ans.     1728. 

3.  What  is  the  volume  of  a  octaedron,  whose  edge  is  20  ? 

A?is.     3771.236. 

4.  What    is    the    volume    of  a    dodecaedron,   whose    edge 
is    25  ?  A71S.     11973G.2328. 

5.  What    is    the    volume    of    an    icosaedron,   whose    edge 
is   20  ?  Ans.     17453.56. 


4 

I 


A  TABLE 


OF 


LOGARITHMS  OF  NUMBERS 


FKOM   1   TO   10,000. 


1 

Log. 

N. 
26 

Log. 

N. 

Log. 

N. 

Log. 

I 

0- 000000 

i -414573 

i-43i364 

5i    I 

707570 

76 

1-880814 

a 

0 

3o!o3o 

3 

52    1 

716003 

77 

I -886491 

3 

0 

477'2i 

;• 447 1 58 

53    1 

724276 

73 

1-892095 

4 

0 

602060 

29 

1-462398 

54    1 

732894 

79 

1-897627 

5 

0 

698970 

3o 

i-477'2i 

55    I 

740363 

80 

I -908090 

6 

0 

778i5i 

3i 

I -491362 

56    I 

748188 

81 

1-908485 

7 

0 

845098 

32 

i-5o5i5o 

ll          \ 

755875 

82 

1-918814 

8 

0 

908090 

33 

i-5i85i4 

763428 

83 

1-919078 

9 

0 

954243 

34 

1-531479 

59    I 

770852 

84 

1-924279 

10 

I 

000000 

35 

1-544068 

60    I 

778151 

85 

1-929419 

11 

041393 

36 

i-5563o3 

61    1 

785330 

86 

1-934498 

12 

079181 

37 
33 

1-568202 

62    1 

792392 

87 

1-939519 
1-944483 

i3 

1 13943 

1-579784 

63    1 

799341 

88 

14 

146128 

39 

I -591065 

64    1 

806181 

89 

1-949390 

ID 

176091 

40 

1-602060 

65    1 

812913 

90 

1-934243 

i6 

204120 

41 

1-612784 

66    1 

819544 

91 

1-959041 
1-963788 

17 

230440 
255273 

42 

1-623249 

u   ; 

826075 

92 

i8 

43 

1-633468 

832509 

93 

1-968483 

19 

278754 

44 

1-643453 

6q    I 

83884Q 

94 

1-978128 

20 

3oio3o 

45 

I-6532I3 

70    I 

845098 

95 

1-977724 

21 

322219 
342423 

46 

1-662758 

7'    I 

851258 

96 

1- 582271 

22 

47 

1-672098 

72    1 

857333 

97 

1-9S677J 

23 

361728 

48 

1-681241 

73    1 

863323 

98 

I -991226 

24 

380211 

49 

1-690196 

74    1 

869282 

99 

1-995635 

25 

1-397940 

5o 

I -698970 

75    1 

875061 

100 

2 • 000000 

Eemark.  In  the  following  table,  in  the  nine  right  hand 
columns  of  each  page,  where  the  first  or  leading  figures 
change  from  9's  to  O's,  points  or  dots  are  introduced  in- 
stead of  the  O's,  to  catch  the  eye,  and  to  indicate  that  from 
thence  the  two  figures  of  the  Logarithm  to  be  taken  from 
the  second  column,  stand  in  the  next  line  below. 


A  TABLE   OF   LOGARITHMS   FROM   1   TO    10,000. 


N. 

0 

I 

2 

3    4 

5 

6 

7 

8 

9 

D.l 

1 

100 

! 

001)000 ! 

0434 

0868 

i3oi   1734 

2166 

2598 

3029 

3461   3891 

43a 

!0I 

432ii 

4751 

5i8i 

5609  6o38 

6466 

6894 

7321 

7748 

8174 

428 

1 02 

8600' 

9026 
3239 

9431 

368o 

9876 

•3oo 

•724 

1147 

1570 

1993 

24i5 

424 

io3 

012837 

4100 

4521 

4940 

536o 

5779 

6197 

6616 

419 

1 04 

7033 

7431 

7868 

8284 

8700  1  9116  1 

9332 
3664 

,947   •36 1  1 

•775 

416 

io5 

021 189  i6o3  1 

2016 

2428 

2841 

3252 

4075 

4486 

4896   412' 

io6 

53  06 

571.5 

6i25 

6533 

6942 

735o 

7757 

8164 

8571  1  S978  1 

4o3 

107 

9384: 

9789 

•195 

•600 

1004 

1408 

1812 

221& 

2619   3021  1 

404 

108 

033424! 

3826 

4227 
8223 

4628 

5029 

543o 

583o 

6230 

6629 

7028 

400 

109 

7426 

7825 

8620 

9017 

9414 

981 1 

•207 

•602 

•998 

396 

no 

041393 

1787 

2182 

2575 

2969   3362 

3755 

4148 

4540 

4932 

393 

III 

53231 

5714 

6io5 

6495 

6883 

7275 

7664 

8o53 

8442 

8830 

3S9 

112 

0218 
05J078 

9606 
3463 

9993 

•38o 

•766 

ii53 

1538 

1924 

23og 

26g4 

386 

ii3 

3846 

423o 

46i3 

4996 

5378 

5760 

6142 

S524 

382 

114 

6903 

7286 

7666 

8046 

8426 

88o5 

9185 

9563 
3333 

9942 

•320 

370 
376 

ii5 

06069S 

1075 

1452 

1829 

2206 

2582 

2q38 

3709 

4o83 

116 

4438 

4832 

5206 

558o 

5953 

6326 

6699 

7071 

7443 

7815 

oP 

"7 

8i86 

8557 

8928 

9298 

9668 
3352 

••38 

•407 

•776 

1145 

i5i4 

^^2 

118 

071882 

225o 

2617 

29S5 

3718 

4o85 

445 1 

4816 

5i82 

366 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

o543 
3i44 

9904 
33o3 

•266 

0626 

•987 
4576 

1 347 

1707 

2067 

2426 

36o 

121 

082783 

3861 

4219 

4934 

52gi 

5647 

6004 

357 

122 

636o 

6716 

7071 

7426 

7781 

8i36 

8490 

8845 

9.98 

9552 
3071 

355 

123 

9905 

•258 

•611 

•963 

i3i5 

1667 

2018 

2370 

2721 

35! 

124 

093422 

3772 

4122 

4471 

4820 

5169 

55i8 

5866 

62i5 

65^2 

^^2 

125 

6qio 

7257 
0715 

7604 

7951 

8298 

8644 

8990 

9335 

9681 
3i  [9 

••26 

346 

126 

1 00371 

loSg 

i4o3 

1747 

2ogi 

2434 

2777 

3462 

343 

III 

38o4 

4146 

4487 

4828 

5169 

55io 

585i 

6191 

6531 

6871 

340 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 
3275 

•253 

338 

129 

iioSgo 

0926 

1263 

1 599 

1934 

2270 

2605 

2940 

3609 

335 

i3o 

1 13943 

4277 

4611 

4944 

5278 

56ii 

5943 

6276 

6608 

6940 

333 

i3i 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

0915 

3198 
6456 

•245 

33o 

l32 

120374 

0903 

I23l 

i56o 

1888 

2216 

2544 

2871 

3525 

328 

i33 

3832 

4178 

45o4 

4S3o 

5i56 

5481 

58o6 

6i3i 

6t8i 

325 

i34 

7103 

7429 

7753 

8076 

8399 

8722 

9045 

9^i^ 

9690 

••12 

323 

i35 

i3o334 

o653 

0977 

1298 

1619 

1939 

2260 

258o 

2900 

3219 
64oi 

321 

1 36 

3539 

3858 

4177 

4496 

4814 

5i33 

545 1 

5769 

6086 

318 

137 

6721 

7037 

7354 

7671 

7987 

83o3 

8618 

8g34 

9249 

9564 

3i5 

i38 

9879 
i43oi5 

•194 

•5o8 

•822 

ii36 

i45o 

1763 

2076 

238g 

2702 

3i4 

139 

3327 

3639 

3951 

4263 

4574 

4885 

5ig6 

5507 

58i8 

3ii 

140 

146128 

6438 

6748 

7o58 

7367 

7676 

7985 

82g4 

86o3 

O911 

3oo 

Wi 

9219 

9527 

9835 

•142 

•449 

•i56 

io63 

1370 

1676 

19S2 

30T 

3o6 

142 

I5228S 

25g4 

2900 

32o5 

35io 

3^i5 

4120 

4424 

4728 

5o32 

!43 

5336 

5640 

5943 

6246 

6549 

6852 

Ti54 

7457 

7759 

8061 

3o3 

144 

8362 

8664 

8965 

9266 

9367 

0868 

•168 

•469 

•760 
373$ 

1068 

3oi 

145 

161 368 

1667 

1967 

2266 

2564 

2863 

3i6i 

3460 

4o55 

299 

146 

4353 

465o 

4947 

5244 

5541 

5838 

6i34 

6430 

6726 

7022 

297 
1  295 

147 

7317 

7613 

7908 

82o3 

8497 

8792 

9086 

g38c 

9674 

'^2  1 

148 

170262 

o555 

0848 

1141 

1434 

1726 

2019 

23ll 

26o3 

2895  ,  29JI 

149 

3i86 

3478 

3769 

4060 

435 1 

4641 

4932 

5222 

55i2 

58o2 

291 

i5o 

1 76091 

638i 

6670 

6959 

7248 

7536 

7825 

?Il3 

8401 

S6«g 

:^89 

i5i 

8077 

9264 

9552 

9^39 

•126 

•4i3 

'Jm 

•g85 
3S.3g 

1272 

1338 

287 

l52 

181 844 

2129 

24i5 

2700 

2985 

3270 

3555 

4123 

4407 

283 

1 53 

4691 

4075 
7803 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

i54 

7321 

8084 

8366 

8647 

8928 

9209 

g490 

9771 

••5i 

28! 

i55 

190332 

0612 

0892 

1171 

1 45 1 

1730 

2010 

2289 

2567  1  jy4fc 

279 

i56 

3.25 

;  34o3 

368 1 

3959 

4237 

45i4 

4792 

5069 

5346   5623 

2-8 

1 59 

5899  6176 
8637,  8932 

6453 

6729 

7oo5 

7281 

7536 

7832 

8107  ,  8332 

276 

9206 

9481 

9755 

••29 

•3o3 

•577 

•85o 

1124 

274 

159 

20139- 

1670 

1943 

2216  1  2488 

2761 

3o33 

33o5 

3577 

3848 

27a 

W. 

0 

I 

a 

3 

4 

' 

6 

1 

7 

8 

9 

D. 

A  TABLE   OF  LOGARITHMS   FKOit   1    TO   10,000. 


N. 

0     I 

2   1  3 
1 

4 

5 

6 

7 

8 

9 

D. 

lOO 

204120  4391 

4663  1  4934 

5:o4 

5475 

5746 

6016 

6286 

6556 

371 

l6i 

163 

6826  7096 

95)5  9783 

7365 
••5 1 

1  7634 
•3 19 

7904 
•586 

8173 
•853 

8441 
1121 

8710 
i388 

'm 

9247 
1921 

4579 

269 

267 

1 63 

212188'  2454 

2720 

2986 

3252 

35i8 

3783 

4049 

43i4 

266 

164 

4844  5109  1  5373 

5638 

5902 

6166 

643o 

6694 

6957 

7221 

264 

i65 

7484!  7747 

8010 

8273 

8536 

8798 

9060 

9323 

9DS5 

9846 

262 

166 

220108'  037a 

o63i 

0892 

ii53 

1414 

1675 

1036 

2196 

2456 

261 

167 

2716  2976 
5309 1  5568 

3236 

3496 

3755 

4oi5 

4274 

4533 

4792 

5o5i 

259 

168 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

763o 

258 

169 

7S87  8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

•193 

256 

170 

230449  0704 

0960 

I2l5 

1470 

1724 

'979 

2234 

2488 

2742 

254 

'71 

2996  325o 

3do4 

3757 

401 1 

4264 

4517 

4770 

5o23 

5276 

253 

172 

5d28 

5781 

6o33 

6285 

6537 

67S9 

7041 

7292 

7544 

7795 

252 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

••5o 

•3oo 

25o 

'H 

24o54q 

0799 

1048 

1207 

3782 

1 546 

1795 

2044 

2293 

2541 

2790 

249 

1-5 

3o3b 

3286 

3534 

4o3o 

4277 

4525 

4772 

5oiq 

5266 

?48 

175 

55i3 

5739 

6006 

6252 

6499 
8954 
1395 

6745 

6991 

7237 

7482 

7728 

246 

177 
178 

7973 

8219 

8464 

8709 

9198 

9443 

9687 

2125 

9032 

2368 

•176 

245 

25o420 

0664 

0908 
3338 

ii5i 

i638 

I88I 

2610 

243 

179 

2853 

3096 

358o 

3822 

4064 

43o6 

4548 

4790 

5o3i 

242 

180 

255273 

55i4 

5755 

Iv^ 

6237 

6477 

6718 

6958 
9355 

7198 

IW^ 

241 

181 

7679 

7918 

8i58 

83q8 

8637 

8877 

9116 

9594 

239 

182 

260071 

o3io 

0548 

0787 

1025 

1263 

i5oi 

1739 

1576 
4346 

22l4 

238 

1 83 

245 1 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4582 

237 

184 

4818 

5o54 

5290 

5525 

6761 

5996 

6232 

6467 

6702 

6937 

235 

i85 

7172 

7406 

7641 

7875 

8II0 

8344 

8578 

8812 

9046 

9279 

234 

186 

95i3 

9746 

9980 

•2l3 

•446 

•679 

•912 

1 144 

'377 

1609 

233 

187 

271842 

2074 

23o6 

2538 

2770 

3ooi 

3233 

3464 

3696 

3927 

232 

188 

4i58 

4389 

4620 

4S5o 

5o8i 

53ii 

5542 

5772 

6002 

6232 

23o 

£89 

6462 

6692 

6921 

7i5i 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

275^754 

8982 

92 1 1 

9439 

9667 

9895 

•123 

•35 1 

•578 

•806 

228 

'91 

281033 

1261 

1488 

1715 

1942 

2169 

23q6 

2622 

2849 

3075 

227 

192 

33oi 

3527 

3753 

J979 

42o5 

443 1 

4656 

4882 

5107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7i3o 

7354 

7578 

225 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

i8i3 

9812 

223 

193 

290035 

0257 

04S0 

0702 

0925 

1147 

1369 

1 591 

2o34 

222 

[96 

2256 

2478 

2699 

2920 

3i4i 

3363 

3584 

38o4 

4025 

4246 

221 

'97 

4466 

4687 

4907 

5i27 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198   6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

8853 

9071 

9289 

9507 

9725 

9943 

•161 

•378 

•595 

•8i3 

2l8 

200 

?oio3o 

1247 

1464 

1681 

1898 

2114 

233i 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5i36 

216 

202 

•535! 

5566 

5781 

5996 

6211 

64:5 

6639 

6854 

70O8 

7282 

2l5 

2o3  1 

7496 

7710 

7924 

8,37 

835 1 

8564 

8778 

8991 

9204 

9417 

2l3 

204  1 

96J0 

9843 

••56 

•268 

•481 

•693 

•906 

ifi8 

i33o 

1 542 

212 

2o5  311754] 

1966 

2177 

4289 

2389 

2600 

281: 

3o23 

3234 

3445 

3656 

211 

206  1  3867, 

4078 

4499 

47'0 

4920 

5i3o 

5340 

555i 

5760 

210 

S07  1 

5970 

6i8o 

6390  6599 

6809 

7018 

7227 

7436 

7646 

7854  1 

209 

208 

8063' 

^272 

8481   8689 

8898 

9106 

93i4 

9522 

9730  1  9938 

208 

209 

33oi46| 

o3d4 

o562  1  0769 

0977 

1 184  i  1391  1 

1598  1 

i8o5  i  2012 

207 

;ic 

322219! 

2426 

2633   2839 

3  046 

3252 

3458 

3665 

3871 

4077  1  206 

an 

4282 

4488 

4694  4899 

5io5 

53io 

55i6 

5721 

5926 

6i3i  1  2o5l 

312 

6336, 

6541 

67^5  6950 

7i55 

735n 

7563 

7767 

7972 

8176  1  204 1 

2l3 

838o' 

8583  ■  8787  1 

8991 

9194 

939S 

9601 

9805 

•••8 

•211 

203 

2!4  t3Jo4i4  0617  1 

0819 

1022 

1225 

1427 

i63o 

1 83  2 

2o34 

2236 

202 

2l5 

2438'  2640 

2842 

3  044 

3246 

3447 

3649 

385o 

4o5i 

4253 

202 

216 

4454:  4655 

4856 

5o57 

5257 

5458 

5658 

5859 

6059 

0260 

201 

l\l 

6460'  6660 

6860  7060 

7260 

7459 

7659 

7858 

8o58 

8257 

200 

8456;  8656 

8855  1  9054 

9253 

9431 

q65o 

9849 

••47 

•246 

D. 

219  . 

340444  0642 

0841   1039 

1237 

1435   i632  { 

i83o 

2028 

2225 

9 

N. 

0 

I 

3 

3141 

5 

6 

7 

»  1 

A  TABLE  OF  LOGAEITHMS  FROM   1   TO   10,000 


N. 

220 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D. 
197 

342423 

2620 

2817 

3oi4 

3212 

3409 

36o6 

38o2 

3999 

4196 

221 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

223 

85o5 

85oo 

8694 

88«9 

9083 

9278 

9472 

9666 

9860 

••54 

194 

224 

350248 

0442 

o636 

0829 

1023 

1216 

1410 

i6o3 

1796 

1989 

193 

2:5 

ii8j 

2375 

2568 

2761 

2954 

3i47 

P^ 

3532 

3724 

3916 

193 

226 

4108 

43oi 

4493 

4685 

48,6 

5o68 

5260 

545: 

5643 

5834 

192 

227 

6026 

6217 
8125 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

223 

7035 
9835 

83i6 

85o6 

8696 

8886 

9076 

9266 

9456 

9646 

190 

229 

••25 

•2l5 

•404 

•593 

•783 

•972 

1161 

i35o 

1 539 

189 

23o 

361728 

1917 

2io5 

2294 

2482 

2671 

2859 

3  048 

3236 

3424 

188 

23 1 

3612 

3800 

3988 

4176 

4363 

455i 

4739 

4926 

5ii3 

53oi 

188 

282 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

233 

7356 

7542 

7729 

791a 

8101 

8287 

8473 

8659 

8845 

9o3o 

186 

234 

9216 

9401 

9587 

9772 

9958 

•143 

•328 

•5i3 

•698 

•883 

i85 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

236o 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

3«3i 

4oi5 

4198 

4382 

4565 

184 

237 

4748 

4932 

5ii5 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

J  83 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8o34 

8216 

182 

239 

8398 

858o 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

••3o 

181 

240 

380211 

0392 

0573 

0754 

0934 

iii5 

1296 

1476 

1 656 

1837 

181 

241 

2017 

2197 

2377 

2D57 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

38i5 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

243 

56o6 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 
88n 

7212 

6989 

178 

544 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

178 

145 

9166 

9343 

9520 

9698 

9875 

••5 1 

•228 

•4o5 

•582 

•759 

177 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

2697 
44^2 

2873 

3043 

3224 

3400 

3575 

37DI 

3926 

4101 

4277 

176 

248 

4627 

4802 

4977 

5i52 

5326 

55oi 

5676 

585o 

6025 

.75 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

25o 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

95oi 

,73 

25l 

9674 

9847 

••20 

•192 

•365 

•538 

•711 

•883 

io56 

1228 

173 

252 

40140: 

1573 

1745 

I9I7 

3635 

2089 

2261 

2433 

26o5 

2777 

2949 

172 

253 

3l2I 

3292 

3464 

3807 

3978 

4149 

4320 

4492 

4663 

171 

254 

4834 

5oo5 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

705 1 
8749 

7221 

7391 

7561 

7731 

7901 

8070 

170 

256 

8240 

8410 

8579 

8918 

9087 

9257 

9426 

9595 

9764 

169 

257 

9933 

•102 

•271 

•440 

•609 

•777 

•946 

1114 

1283 

1451 

169 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3i32 

168 

259 

33oo 

3467 

3635 

38o3 

3970 

•4i37 

43o5 

4472 

4639 

4806 

167 

260 

414973 

5i4o 

5307 

5474 

5641 

58o8 

5974 

6141 

63o8 

6474 

167 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8i35 

166 

262 

83oi 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 
1439 

i65 

263 

9956 

•121 

•286 

•45 1 

•616 

•781 

•945 

IIIO 

1275 

i65 

264 

421604 

1788 

1933 

2007 
3737 

2261 

2426 

2390 

2754 

2918 

3082 

164 

265 

3246 

341C 

3574 

3901 

4o65 

4228 

4392 

4555 

4718 

164 

266 

4882 

5045 

5208 

5371 

5d34 

5697 

586o 

6023 

6186 

6340 

7973 

1 63 

267 

65ii 

6674 

6836 

6999 

7161 

8783 

7324 

7486 

7648 

781 1 

.'62 

268 

8i35 

8297 

8459 

8621 

8944 

9106 

9268 

9429 

9391 

163 

J69 

9752 

9914 

••7? 

•236 

•398 

•559 

•720 

•881 

1042 

I203 

ibi 

270 

43i364 

i525 

i685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3i3o 

3290 

3430 

36io 

3770 

3930 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5d26 

5685 

5844 

6004 

1 59 

2/3 

6i63 

6322 

6481 

6640 

6798 

6957 

7116 

■7275 
8859 

7433 

7592 

i5g 
1 58 

274 

775i 

7909 

8067 

8226 

8384 

8542 

8701 

9017 

9175 

275 

9333 

9491 

9648 

q8o6 

Uti 

•122 

"279 

•437 

•594 

•752 

1 58 

276 

440909 

1066 

1224 

iJ'^'. 

1695 

i85i 

^009 

2166 

2323 

1 57 

277 

2480 

2637 

2793 

2950 

3io6 

3263 

3419 

3576 

^73? 

3889 

■  57 

278 

4045 

4201 

4357 

45i3 

a '-69 

4825 

49«i 

5i37 

5293 

5449 

1 56 

279 

56o4 

5760 

5915 

6071 

622c 

6382 

6537 

6692 

6848 

7003 

i55 

N. 

e 

I 

2 

3 

4 

3 

6 

7 

8 

9 

1). 

A  TABLE  OF  LOGARITHMS   FROM  1   TO  10,000. 


N.  :  0 

I 

2 

3 

4 

5 

6     7 

8 

9 

D. 

1 55 

28o 

447158 

73i3 

7468 

7623 

7778 

7933 

8088 

8242 

6397 

8552 

j8i 

8706 

8861 

90i5 

9170  1  9324 

9478 

9633 

9787 

9941 

••nS 

1 54 

j8j 

450249 

o4o3 

0557 

071 1 

o865 

1018 

1172 

i326 

1479 

1 633 

1 54 

283 

1 786 

1940 

2093 

2247 

2400 

2553 

2706 

2839 

3oi2 

3i65 

i53 

284 

33i8 

3471 

3624  '   3777 

3930 

4082 

4235 

4387 

4540 

4692 

i53 

:  385 

4845 

4997 

5i5o  !  53o2 

5454 

56o6 

5758 

5910 

6062 

6214 

l53 

n86 

6366 

65i8 

6670  1  6821 

6973 

7125 

7276 

7428 
8940 

7579 

773. 

l52 

287 

7S82 

8o33 

81 84 

8336 

8487 

8638 

8789 

9091 

9242 

i5i 

28S 

9392 

9543 

9694 

9S45 

9993 

•146 

•296 

•447 

•597 
2098 

•748 

i5i 

289 

460S98 

1048 

1 198 

1348 

1499 

1649 

1799 

1948 

J248 

i5o 

290 

462398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

i5o 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5oS5 

3234 

149 

292 

5383 

5532 

568o 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8o52 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

295 

9822 

9969 

•116 

•263 

•410 

•557 

•704 

•85i 

•998 

1145 

147 

296 

471292 

1438 

1 585 

1732 

1878 

2025 

2171 

23i8 

2464 

2610 

146 

297 

2736 

2903 

3o49 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 1 

298 

4216 

4362 

45o8 

4653 

4799 

4944 

5090 

5235 

538i 

5526 

146 

299 

5671 

58i6 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

3oo 

477121 

7266 

741 1 

7555 

7700 

7844 

7989 

8i33 

8278 

8422 

145 

3oi 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

3o2 

48oooT 

oi5i 

0294 

0438 

o582 

072D 

0869 

1012 

1136 

1299 

144 

3o3 

1443 

i586 

1729 

1872 

2016 

2159 

23o2 

2445 

2588 

2731 

143 

3o4 

2874 

3oi6 

3i59 

33o2 

3445 

3587 

3730 

3872 

4oi5 

4157 

143 

3o5 

43  00 

4442 

4585 

4727 

4869 

Sou 

5i53 

5295 

5437 

5579 

142 

3o6 

5721 

5863 

6oo5 

6147 

6289 

643o 

6572 

6714 

6855 

6997 

142 

3o7 

7i38 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

3o8 

855i 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

809 

9958 

••99 

•239 

•38o 

•520 

•661  •  ^80 1 

•941 

1081 

1222 

140 

3io 

491362 

i5o2 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

3ii 

2760 

2900 

3040 

3179 

3319 

3458 

35q7 

3737 

3876 

4oi5 

139 

3l2 

4i55 

4294 

4433 

4572 

4711 

485o 

4989 

5128 

5267 

5406 

'39 

3i3 

5544 

5683 

5822 

5960 

6099 
74S3 

6238 

6376 

65i5 

6653 

6791 

139 

3i4 

6g3o 
83 1 1 

7068 

7206 

1344 
8724 

7621 

7759 

7897 

8o35 

8173 

i38 

3i5 

8448 

8586 

8862 

8999 

9137 

9275 

9412 

9550 

i38 

3i6 

9687 

9824 

9962 

••99 

•236 

•374 

•5ii 

•648 

•785 

•922 

137 

3i7 

5oio59 

1 196 

1333 

1470 

1607 

1744 

1880 

2017 

2i54 

2291 

'^7 

3i8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

35i8 

3635 

i36 

319 

3791 

3927 

4o63 

4199 

4335 

4471 

4607 

4743 

4878 

5oi4 

i36 

320 

5o5i5o 

5286 

5421 

5557 

5693 

5828 

5o64 
7316 

6099 

6234 

6870 

i36 

321 

65o5 

6640 

6776 

6911 

7046 

7181 

7431 

7586 

7721 

i35 

322 

7856 

7991 

8126 

8260 

8395 

853o 

8664 

8799 

8934 

9068 

i35 

323 

9203 

9337 

9471 

9606 

9740 

9874 

•••9 

•143 

•277 

•411 

i34 

1  324 

5io545 

0679 

08 1 3 

0947 

1081 

I2l5 

1 3  49 

1482 

1616 

1730 

1 34 

325 

1 883 

2017 

2l5l 

2284 

2418 

255i 

2684 

2818 

2931 

3o84 

i33 

326 

3218 

335i 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

i33 

327 

4548 

4681 

48i3 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

i33 

323 

5874 

6006 

6139 

6271 

64o3 

6535 

6668 

6800 

6932 

7064 

8382 

l32 

329 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

825i 

l32 

33o 

5i85i4 

8646 

8777 

8909 

9040 

9171 

93o3 

9434 

9566 

9697 

i3i 

33i 

9828 

9959 

••90 

•221 

•353 

•484 

•6i5 

•745 

•876 

1007 

i3i 

332 

S2II38 

1269 

1400 

i53o 

1661 

1792 

1922 

2033 

2i83 

23i4 

i3i 

333 

2444 

257D 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

36i6 

i3o 

334 

3746 

3876 

4006 

4i36 

4266 

4396 

4526 

4656 

4785 

49' 3 

i3o 

335 

5o45 

5i74 

53o4 

5434 

5563 

5693 

5822   5951 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114  7243 

7372 

7501 

129 

337 
338 

7630  775g 

7888 

8016 

8145 

8274 

8402   853 1 

8660 

8788 

139 

8917  9045 

9'74 

930J 

943o 

9559 

9687 

9815 

9943 

••72 

ia8 

339 

530300  o328 

0456 

o584 

0712 

0840  0968 

1096 

1333 

8 

i35i 

138 
D. 

■  N. 

0 

I 

a 

3 

4 

5 

6 

7 

9 

6 


A  TABLE   OF   LOGARITHMS  FROM   1   TO   10,000. 


N. 

0    I   1  2 

3     4   1   5 

,  1 

6 

7 

8 

9 

D. 

340 

531479  1607 

1734 

1862 

1990  21 17 

2245 

2872 

25oo 

2627 

128 

3H 

2734  .2882 

3009 

3i36 

3264 

3891 

35i8 

3645 

8772 

3899 

127 

342 

4026  41 53 

4280 

4407 

4584 

4661 

4787 

4914 

5o4i 

5167 

127 

343 

5294  5421 

5547 

5674 

58oo 

5927 

6o53 

6180 

63o6 

6482 

126 

344 

6558  66S5 

6811 

6937 

7068 

7189 
8448 

7815 

7441 

& 

25s 

126 

3/|J 

7819  7945 

8071 

8197 

8322 

8574 

8699 

126 

34& 

90761  9202 

9327 

9452 

9578 

9708 

9829 

9954 

••79 

•204 

125 

347 

540829  0455 

o58o 

0705 

o33o 

0955 

1080 

i2o5 

i83o 

1454 

125 

34^ 

1579:  1704 

1829 

1953 

2073 

2208 

2827 

2452 

2576 

2701 

125 

349 

28i5|  2950 

3074 

3199 

3328 

3447 

3571 

8696 

3820 

3944 

124 

35o 

544068J  4192 

43i6 

4440 

4564 

4688 

4812 

4986 

5o6o 

5i88 

124 

35i 

5307 1  543 1 

5555 

5678 

58o2 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543'  6666 

6789 

6913 

7086 

7159 

7282 

74o5 
8635 

7529 
6753 

7632 

123 

353 

7775  7«98 

8021 

8144 

8267 

88S9 

85i2 

8881 

123 

354 

9003  9126 

9249 

9871 

9494 

9616 

9789 

9861 

9984 

•106 

128 

355 

350228'  o35i 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

i45o  1572 

1694 

i8i6 

1988 

2060 

2181 

2808 

2425 

2547 

122 

l^l 

2668 

2790 

29H 

3o33 

3i55 

3276 
4489 

3398 

35i9 

3640 

8762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4610 

4781 

4852 

4978 

121 

359 

5094 

52i5 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

5563o3 

6423 

6544 

6664 

6785 

6905 

7026 

7146 
8849 

7267 
8469 

7887 

120 

36 1 

7307 

7627 

7748 

7868 

7988 

8io3 

8228 

8589 

120 

362 

8709 

8829 

8943 

9068 

9188 

9808 

9428 

9548 

9667 

9787 

120 

363 

9907 

••26 

•146 

•265 

•885 

•5o4 

•624 

•743 

•863 

•982 

119 

364 

56IIOI 

1221 

1 340 

1459 

1578 

1698 

1817 

1986 

2o55 

2174 

119 

365 

2293 

2412 

253i 

265o 

2769 

2887 

3  006 

3i25 

3244 

3862 

119 

366 

3481 

36oo 

3718 

3837 

8955 

4074 

4192 

43ii 

4429 

4548 

119 

367 

4666 

4784 

4903 

5o2i 

5i39 

5257 

5376 

5494 

56i2 

5780 

118 

568 

5848 

5966 

6084 

6202 

6320 

6487 

6555 

6673 

6791 

6909 

118 

J69 

7026 

7144 

7262 

7379 

7497 

7614 

7782 

7849 

7967 

8oa4 

118 

»70 

568202 

83i9 

8436 

8554 

8671 

8788 

8905 

9028 

9140 

9257 

117 

(71 

9374 

9491 

9608 

9725 

9842 

9959 

••76 

•193 

•809 

•426 

117 

!^i 

570543 

0660 

0776 

0893 

lOIO 

1126 

1243 

1839 

1476 

1592 

"7 

73 

1709 

1825 

1942 

2o58 

2174 

2291 

2407 

2523 

2689 

2735 

lit. 

^4 

2872 

2988 

3io4 

3220 

8336 

3452 

3568 

3684 

3800 

8915 

116 

^7^ 

4o3i 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5i88 

53o3 

5419 

5534 

565o 

5765 

588o 

5996 

6111 

6226 

ii5 

^A 

6341 

6457 

6572 

6687 

6802 

6917 

7082 

7147 

7262 

nil 

ii5 

378 

7492 

7607 

7722 
8863 

7836 

7951 

8066 

8181 

8295 

8410 

ii5 

379 

8639 

8754 

8983 

9097 

9212 

9826 

9441 

9555 

9669 

114 

380 

579784 

9898 

•0,2 

•126 

•241 

•855 

•469 

•583 

•697 

•811 

114 

38i 

580925 

1089 

ii53 

1267 

i38i 

1495 

1608 

1722 

1 836 

1930 

114 

382 

2o63 

2177 

2291 

2404 

25i8 

2681 

2745 

2858 

2972 

3o85 

114 

383 

3i99 

33i2 

3426 

3539 

3652 

8765 
4896 

3879 

8992 

4io5 

4218 

ii3 

384 

433 1 

4444 

4557 

4670 

4788 

5oo9 

5l22 

5285 

5348 

ii3 

385 

5461 

5574 

5686 

5799 

5912 

6024 

6187 

6250 

6862 

6475 

ii3 

386 

6587 

6700 

6812 

692D 

7087 

7149 

7262 
8884 

7374 

8496 

7486 
8608 

7599 

112 

387 

77" 
8832 

7823 
8944 

7935 

8047 

8160 

8272 

8720 

112 

388 

go56 

9167 

9279 

9891 

95o3 

9615 

9726 

9888 

112 

389 

9950 

••6i 

•173 

•284 

•896 

•5o7 

•619 

•780 

•842 

•953 

112 

390 

591065 

1176 

1287 

1899 

i5io 

if  31 

1782 

1843 

1955 

2066 

III 

39! 

2177 

2288 

2399 

25lO 

2621 

2782 

2843 

2954 

3064 

3.75 

III 

M 

3286 

3397 

35o8 

36i8 

8729 

3840 

8950 

4061 

4171 

4282 

III 

393 

4393 

45o3 

4614 

4724 

4834 

4945 

5o55 

5 1 65 

5276 

5J86 

110 

M 

5496;  56o6 

5717 

5827 

5987 

6047 

6,57 

6267 

6377 
7476 
8072 

64S7 

no 

^A 

6597  6707 

6817 

6927 

1087 

7146 
8243 

7256 

7866 

7586 

no 

396 

7695  7805 

7914 

8024 

8184 

8853 

8462 

8681 

no 

397 

B791 
9883 

8900 

9009 

9^9 

9228 

9337 

9446 

9556 

9665 

9774 

109 

39S 

9992 

•lOI 

•210 

•819 

•428 

•537 

•646 

•755 

•S64 

109 

399 

600973 

1082   1191 

1299 

1408 

i5i7 

1625 

1734 

1843 

1931 

109 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

A  TABLE   OF   LOdARITHMS   FROM   1   TO   10,000. 


.■N. 

u 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

108 

400 

602060 

3203 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3o36 

401 

3i44 

3361 

3469 

3577 

3686 

3704 

3902 

4010 

4118 

108 

402 

4226 

4334 

4442 

455o 

4658 

4766 

4874 

4982 

5089 

5197 

108 

4o3 

53o5 

541 3 

5521 

5628 

5i36 

5844 

5951 

6039 

6166 

6274 

108 

/.q4 

6381 

6489 

6596 

6704 

681 1 

6919 

7026 

7i33 
6205 

7241 

g3l2 

7348 
8419 

107 

4o3 

7453 

7062 

7669 

7777 

7884 

7991 

S098 

107 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9881 

9488 

107 

407 

9)94 

9701 

9808 

9914 

••21 

•128 

•234 

•341 

•447 

0554 

107 

40& 

6 1 0660 

0767 
1829 

0873 

0979 

1086 

1192 

1298 

i4o5 

i5ii 

1617 

106 

409 

1723 

1936 

2042 

2148 

2254 

236o 

2466 

2572 

^678 

106 

.4:0 

(12784 

2S90 

2996 

3l02 

3207 

33i3 

3419 

3525 

363o 

3736 

106 

•in 

3842 

3947 

4oD3 

4139 

4264 

4370 

4475 

458 1 

4686 

4792 

106 

412 

4897 

5oo3 

5io8 

521 3 

5319 

5424 

552g 

5634 

5740 

5843 

io5 

4i3 

DgDO 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

6790 

6895 

io5 

414 

7000 

7io5 

7210 

73i5 

7420 

7525 

7629 

7734 

7839 

7943 

103 

41 5 

8048 

8i53 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

io5 

416 

9093 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

••32 

104 

417 

620 1 J6 

0240 

o344 

0448 

o552 

o656 

0760 

0864 

0968 

1072 

104 

41S 

1176 

1280 

i384 

1488 

1592 

1695 

•799 

1903 

2007 

2II0 

104 

419 

2214 

23i8 

2421 

2525 

2628 

2732 

2835 

2939 

3o42 

3146 

104 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

io3 

421 

4282 

4385 

4488 

4391 

4695 

4798 

4901 

5004 

5107 

5210 

io3 

422 

53i2 

541 5 

55i8 

5621 

5724 

5827 

5929 

6o3-2 

6i35 

6238 

io3 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7o38 

7161 

8i85 

7263 

io3 

424 

7366 
8389 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8287 

102 

423 

8491 

8093 

8695 

S797 

8900 

9002 

9104 

9206 

g3o8 

102 

426 

9410 

95 12 

9613 

97i5 

9817 

9919 

o«2i 

•123 

•224 

0326 

102 

427 

630428 

o53o 

o63i 

0733 

oS35 

0936 

io33 

1139 

1241 

1 342 

102 

428 

1444 

1 545 

1647 

I748» 

1849 

1951 

2052 

2i53 

2233 

2356 

lOI 

429 

2457 

2559 

2660 

2761 

2802 

2963 

3064 

3i65 

3266 

3367 

lOI 

43o 

633468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

100 

43 1 

4477 

4578 

4679 

4779 

4880 

4981 

5o8i 

5182 

5283 

5383 

100 

432 

5484 

5584 

56b3 

5785 

5886 

5986 

60S7 

6187 

6287 

6388 

100 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 
8190 

7290 

JlT 

100 

434 

7490 

7590 

7690 

7790 

2^9° 

7990 

8090 

8290 

8389 

99 

435 

8489 

8589 

86»9 

8789 

88«8 

8988 

9088 

9188 

9287 

9387 

99| 

436 

9.IS6 

9586 

9686 

9785 

9885 

9984 

••84 

»i83 

•283 

•382 

99 

437 

640481 

o58i 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

43s 

1474 

1573 

1672 

1771 

187I 

1970 

2069 

2168 

^^=1 

2366 

99 

439 

2  465 

2563 

2662 

2761 

2860 

2959 

3o58 

3i56 

3255 

3354 

99 

440 

643453 

355 1 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

98 

441 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

98 

442 

5422 

5521 

5619 

5717 

58i5 

5913 

60U 

6110 

6208 

63o6 

98 

443 

6404 

65o2 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

9^ 

444 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8i65 

8362 

98 

445 

836o 

8458 

8555 

8653 

8750 

8848 

8943 

9043 

9140 

9237 

97 

446 

9335 

9432 

953o 

9627 

9724 

9821 

9919 

••16 

oii3 

•210 

97 

447 
443 

65o3o8 

o4o5 

0502 

0599 

0696 

0793 

o8qo 

0987 

1084 

1181 

97 

1278 

1375 

1472 

1 569 

i6bO 

1762 

1809 

1956 

2o53 

2i5o 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3ii6 

97 

451 

653213 

3309 

34o5 

35o2 

3598 

8695 

3791 

3888 

3984 

4080 

96 

4T' 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

485o 

4946 

3042 

9^ 

452 

5 1 38 

5235 

533 1 

5427 

5523 

5619 

5715 

58io 

5906 

6002 

96 

453 

6098 

6104 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

9^ 

45.1 

7o56  71D2 

7247 

7343 

7438 
8393 

7534 

7629 

7723 

7820 
8774 

7916 
8870 

9^ 

455 

801 1 1  8107 

8202 

8298 

8488 

8584 

8679 

95 

456 

8965 

9060 

9155 

9250 

9346 

9441 

9336 

9631 

9726 

9821 

95 

457 

9916 

••11 

•106 

•201 

•296 

•391 

•48b 

•58i 

•676 

'^^l 

93 

458 

66oS65 

0060 

io55 

ii5o 

1245 

i339 

1434 

;529 

1623 

1718 

9? 

459 

i8i3  1907 

2002 

2096 

2191 

2280 

238o 

2473 

2369 

2663 

95 

"■ 

0     I 

a 

3 

4 

5 

6 

7 

8 

1 

9 

1 . 

A.  TABLE   OF   LOGARITHMS   FROM   1 


TO   10,000 


N. 
460 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

r ' 

D. 

662755 

2852 

294-) 

3o4i 

3i35 

323o 

3324 

3418 

35i2 

3607 

454§ 

94 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

94 

462 

4642 

4736 

483o 

4924 

5oi8 

5lI2 

5206 

5299 

5393 

5487 

94 

463 

558 1 

5675 

5769 

5862 

5956 

6o5o 

6143 

6237 

633 1 

6424 

94 

464 

65i8 

6612 

6705 

6799 

6892 

6986 

7079 
8oi3 

7173 

7266 

7360 

94 

465 

7453 
^386 

7546 

7640 
8572 

7733 

7826 

7920 

8106 

8199 

8293 

93 

466 

8479 

8665 

8759 

8852 

8945 

9o38 

9i3i 

9224 

93 

467 
466 

9317 

9410 

95o3 

9596 

9689 

9782 

9875 

9967 

••60 

•i53 

93 

670246 

0339 

043 1 

0324 

0617 

0710 

0802 

0895 

09S8 

1080 

93 

469 

1173 

126J 

i358 

1431 

i543 

i636 

1728 

1821 

1913 

20o5 

93 

470 

672098 

2190 

2283 

2373 

2467 

2  56o 

2652 

2744 

2836 

2929 

92 

471 

302I 

3ii3 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

3830 

95 

472 

3942 

4o34 

4126 

4218 

43 10 

4402 

4494 

4586 

4677 

4769 

92 

473 

4861 

4953 

5o45 

5i37 

5228 

5320 

5412 

55o3 

5593 

5687 

92 

474 

5778 

5870 

5962 

6033 

6145 

6236 

6328 

6419 

65i. 

6602 

92 

475 

6694 

6785 

6876 

6968 

7059 

7i5i 

7242 

7333 

7424 
8336 

7516 

91 

476 

7607 

7698 

7789 

7881 

7972 

8o63 

8i54 

8245 

8427 

9« 

477 

85i8 

8609 

8700 

879, 

8882 

8973 

9064 

9135 

9246 

9337 

9' 

47^ 

942S 

9519 

9610 

9700 

979' 

9882 

9973 

••63 

•i54 

•245 

9' 

479 

68o336 

0426 

o5i7 

0607 

0698 

0789 

0879 

0970 

1060 

II3I 

9' 

4S0 

681241 

i332 

1422 

i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

90 

43 1 

2145 

2235 

2326 

2416 

2  3o6 

2596 

2686 

2777 

2867 

2937 

90 

482 

3o47 

3i37 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4933 

5o25 

5i  14 

3204 

5294 

5383 

5473 

5563 

5632 

% 

485 

5742 

5j3i 

5921 
68i5 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

486 

6636 

6726 

6904 

6994 

7o83 

7172 

nnf\i 
1  -'^" 

735i 

7440 

89 

487 

7529 
8420 

7618 

7707 

7796 

7886 

7975 

8064 

8!53 

8242 

833 1 

89 

488 

85o9 

8398 

8687 

8776 

8863 

8qj3 

9042 

9i3i 

9220 

89 

489 

9309 

9398 

9486 

9375 

9664 

9753 

9841 

9930 

••19 

•107 

89 

490 

690106 
1081 

0285 

0373 

0462 

o55o 

0639 

0718 

0816 

0905 

0993 

ll 

491 

1170 

1258 

1347 

1435 

i524 

1612 

1700 

1789 

1877 

4q2 

1965 

2o53 

2142 

223o 

23i8 

2406 

2494 

2583 

2671 
355i 

2759 

88 

493 

2847 

2935 

3o23 

3iii 

3199 

3287 

3375 

3463 

3639 

■88 

494 

3727 

38i5 

3903 

3991 

4078 

4i65 

4254 

4342 

443o 

4517 

88 

495 

4605 

469  3 

4781 

48(^8 

4956 

5o44 

5i3i 

5219 

5307 

5394 

88 

496 

5482 

55t>9 

5637 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

6444 

653i 

6618 

6706 

6793 

6880 

6968 

7o55 

7142 
8014 

87 

498 

7229 

73.7 

7404 

7491 

7578 
8449 

7665 

7732 

7^39 
8709 

7926 

87 

499 

8ioi 

8188 

8275 

8302 

8535 

8622 

8796 

8883 

87 

5oo 

698970 

9057 

9144 

923. 

9317 

94o4 

0491 

9578 

9664 

975i 

87 

5oi 

9S38 

9924 

•  9,1 

••98 

•184 

"271 

«338 

0444 

•53 1 

•617 

87 

5o2 

700704 

0790 

0877 

0963 

io5o 

ii36 

1232 

i3o9 

1395 

1482 

86 

5o3 

i568 

1634 

I  74  I 

1827 

1913 

1999 

2086 

2172 

2238 

2344 

86 

5o4 

243 1 

25i7 

2603 

2689 

2775 

2861 

2947 

3o33 

3ii9 

32o5 

86 

5o5 

3291 

3377 

3463 

3549 

3635 

3721 

38o7 

3893 

3979 

406  5 

86 

rK)6 

4IDI 

4236 

4322 

4408 

4494 

4579 

4665 

4731 

4837 

4922 

86 

507 

5oo8 

5094 

5179 

5263 

5350 

5436 

5522 

56o7 

5693 

5778 

86 

5oS 

5864 

5q49 

6o35 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

609 

6718 

68o3 

6838 

6974 

7059 

7144 

7229 

73i5 

7400 

■'485 

85 

5io 

707570 

7655 

7740 

7826 

79" 

7996 

8081 

8166 

825i 

8336 

85 

5ii 

8421 

85o6 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

5l2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

••33 

85 

5i3 

710117 

0202 

0287 

0371 

0436 

0340 

0623 

0710 

0791 

0879 

85 

5i4 

0963 

1048 

ll32 

1217 

i3oi 

1 385 

1470 

i554 

1639 

1723 

84 

5:5 

1807 

1892 

1976 
2818 

2060 

2144 

2229 

23.3 

2397 

2481 

2566 

84 

Dlt 

265o 

2734 

2902 

2986 

3070 

3i54 

3238 

3323 

3407 

84 

5'I 

5id 

349' 

3575 

3i)5g 

5742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

433o 

4414 

4497 

458i 

4665 

4749 

4833 

4916 

5  000 

5o84 

84 

5i9 

5167 

525i 

5335 

5418 

55o2 

5586 

5669 

5753 

5836 

5930 

84 
D. 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

A  TABLE   OF   LOGARITHMS   FROM    1   TO    10,000. 


N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 
6671 

1   9 
6754 

D. 

83 

520 

716003 

6087 

6170 

6254 

6337 

6421 

65o4 

6588 

5a  I 

6838 

6921 

7004 

7088 

7171 
8oo3 

7254 

7338 

7421 

7504 

7587 

B3 

523 

7671 
8do2 

7754 

8668 

7920 

8086 

8169 

8253 

8336 

8419 

83 

523 

8585 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

52i 

9331 

9414 

9497 

9380 

9663 

9743 

9828 

991 1 

9994 

••77 

83 

523 

720159 

0242 

o325 

040- 

123o 

0490 

0573 

o655 

0738 

0821 

3903 

83 

526 

ooStj 

1068 

ii5i 

i3i6 

1398 

1481 

1 563 

1646 

1728 

82 

527 

5i8 

1811 

1893 

1975 

2038 

2140 

2222 

23o5 

2387 

2469 

2552 

82 

2634 

2716 

2798 

2881 

2963 

3o45 

3127 

3209 

3291 

3374 

82, 

529 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4o3o 

4112 

4194 

82 

53o 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5oi3 

82 

53 1 

5093 

5176 

5238 

5340 

5422 

55o3 

5585 

5667 

5748 

583o 

82 

532 

5912 

5993 

6075 

6i56 

6238 

6320 

6401 

6483 

6564 

6646 

83 

533 

6727 

6B09 
7623 

8435 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

534 

7341 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

535 

8354 

85i6 

8397 

8678 

8739 

8841 

8922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

537 

9974 

••55 

•i36 

•217 

•298 

•378 

•459 

•540 

•621 

•702 

81 

538 

730782 

o863 

0944 

1024 

iio5 

1186 

1266 

1 347 

1428 

i5o8 

81 

539 

1589 

1669 

1750 

i83o 

1911 

1991 

2072 

2l52 

2233 

23i3 

81 

540 

732394 

2474 

2555 

2635 

2715 

2796 

2876 

2936 

3o37 

3117 

80 

541 

3197 

3278 

3358 

3438 

35i8 

3598 

3679 

3739 

3839 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320 

4400 

44«o 

456o 

4640 

4720 

80 

543 

4800 

4880 

4960 

5o4o 

5l20 

5200 

5279 

5359 

5439 

55i9 

80 

544 

5599 

5679 

5739 

5838 

5918 

5998 

6078 

6137 

6237 

63i7 

80 

545 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6934 

7034 

71.3 

80 

546 

7193 

7272 
^067 

7352 

7431 

7511 
83o5 

7590 

7670 

7829 
8622 

7908 

79 

547 

79«7 
87S1 

8146 

8225 

8384 

8463 

8701 

79 

548 

8860 

8939 

9018 

9097 

9177 

9236 

9335 

9414 

9493 

79 

549 

9572 

965i 

9731 

9810 

9S89 

9968 

•047 

•  126 

•2o5 

•284 

79 

55o 

740363 

0442 

052I 

0600 

0678 

0757 

o836 

0915 

0994 

1073 

79 

55i 

Il52 

I23o 

1 309 

1 383 

1467 

1546 

1624 

1703 

1782 

i860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

241 1 

2489 

2568 

2647 

S 

553 

2723 

2804 

2882 

2961 

3o39 

3iiS 

3196 

3273 

3353 

343 1 

554 

35io 

3588 

3667 

3745 

3823 

3902 

3980 

4038 

4i36 

42i5 

78 

555 

4293 

4371 

4449 

4328 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5073 

5i53 

D23l 

53o9 

5387 

5465 

5543 

5621 

5699 

3777 

78 

557 

5855 

5933 

601 1 

6089 

6167 

6243 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7'79 

7256 

7334 

78 

559 

74i2 

7489 

7367 

7645 

7722 

7800 

7878 

7933 

8o33 

8110 

78 

56o 

748188 

8266 

8343 

8421 

849S 

8576 

8653 

8731 

8808 

8885 

77 

56i 

8963 

9040 

91.8 

9193 

9272 

9350 

9427 

9504 

9582 

9659 

77 

562 

9736 

9814 

9891 

9968 

••43 

•123 

•200 

•277 
1048 

•354 

•43 1 

77 

563 

75o3o8 

o586 

0663 

0740 

0817 

0894 

0971 

II25 

1202 

77 

564 

1279 

i356 

1433 

i5io 

1 587 

1664 

1741 

1818 

1895 

1972 

77 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2309 

2586 

2663 

2740 

77 

566 

2816 

2893 

2970 

3o47 

3i23 

3200 

3277 

3353 

343o 

35o6 

77 

567 
568 

3583 

366o 

3736 

38i3 

3889 

3966 

4042 

4119 

4883 

4195 

4272 

]l 

4348 

4425 

45oi 

4578 

4654 

4730 

4807 

4960 

5o36 

569 

5lI2 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

753875 

5951 

6027 

6io3 

6180 

6256 

6332 

6408 

6484 

656o 

76 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 
8oo3 

7320 

8079 

76 

572 

8l33 

7472 
8230 

l^^f 

7624 

7700 

7775 

7831 

7927 

76 

573 

83o6 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

574 

■8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

95,7 

9592 

76 

575 

9668 

9743 

9819 

9894 

9970 

••45 

•121 

•196 

•272 

•347 

It 

576 

760422 

0498 

0373 

0649 

0724 

0799 

0875 

0930 

1025 

IIOI 

75 

^^2 
57S 

1176 

I25l 

i326 

1402 

1477 

l532 

1627 
237^ 

1702 

1778 

i853 

75 

1928 

2oo3 

2078 

2i53 

2228 

23o3 

2453 

2529 

2604 

75 

579 

J679 

2754 

2829 

2904 

2978 

3o53 

3ii8 

32o3 

327a 

3353 

75 

0 

I 

3 

3 

4 

5 

1 

6 

7 

8 

9 

D. 

1 

10 


A   TAtLS    OF   LOGARITHMS   FROil   1    TO    10,000. 


58o 

0 

1 

35o3 

2 

3  1  4 

5 

6 

7 

8  1   9 

1). 

763428 

3578 

3653 

3727 

38o2 

3877 

3952 

4027 

4101 

75 

58 1 

4176 

4231 

4326 

4400 

4475 

4530 

4624 

4699 

•4774 

4848 

75 

582 

4923 

499"^ 

5072 

5i47 

5221 

5296 

5370 

5445 

5320 

5594 

75 

583 

5669 

3743 

58i8 

5892 

5966 

6041 

6ii5 

6190 
6933 

6264 

6^38 

74 

584 

6413 

6487 

6562 

6636 

6710 

6783 

6839 

7007 

7082 

74 

585 

7i56 

7230 

7304 

7379 

7453 

7327 

7601 

7675  ;  7749 

7823 

74 

586 

7S98 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490  ]  8364 

74 

587 

8638 

8712 

8786 

8860 

8984 

9008 

9082 

91 56 

9280 

9803 

74 

588 

9377 

9431 

9525 

9599 

9673 

9746 

9820 

9394 

9968 

••42 

74 

589 

7701 i5 

0189 

0263 

o836 

0410 

0484 

0557 

0681 

0705 

0778 

74 

5qo 

770S52 

0926 

0999 

1073 

n46 

1220 

1298 

1867 

1440 

i5i4 

74 

59, 

i587 

it>6i 

1784 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

592 

2322 

2395 

2468 

2542 

26i5 

2688 

2762 

2885 

2908 

2981 

73 

593 

3oi5 

3128 

3201 

3274 

3348 

342  1 

3494 

3567 

3640  1  3713 

73 

594 

3786 

386o 

8983 

4006 

4079 

4132 

4225 

4298 

4371 

4444 

73 

595 

4517 

4590 

4663 

4786 

4809 

4S82 

4955 

5o28 

5ioo 

5178 

73 

596 

5246 

5319 

5892 

5465 

5538 

56io 

5688 

5756 

6829 

5902 

73 

397 

5974 

6047 

6120 

6198 

6265 

6338 

6411 

6488 

6556 

6629 

73 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7854 

73 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778101 

8224 

82  96 

8368 

8441 

85i3 

8585 

8658 

8780 

8802 

72 

601 

8874 

8947 

9019 

9091 

9168 

9286 

9808 

9880 

9452 

9524 

72 

602 

9396 

9669 

9741 

9813 

9885 

99  S  7 

•»29 

•lOl 

•173 

•245 

72 

6o3 

780317 

0889 

0461 

o583 

o6o5 

0677 

0749 

0S21 

0893 

0963 

72 

604 

1087 

1109 

1181 

1253 

1824 

1896 

1468 

1540 

1612 

1684 

72 

6o5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2829 

2401 

72 

606 

2473 

2544 

2616 

2688 

2739 

2881 

2902 

2974 

8040 

8117 

72 

607 

3189 

3260 

3332 

8408 

3475 

3346 

36i8 

8689 

8761 

3832 

71 

608 

3go4 

3975 

4046 

4118 

4189 

4261 

4332 

44o3 

4473 

4546 

7' 

609 

4(Ji7 

4089 

4760 

4881 

4902 

4974 

5o45 

5ii6 

5187 

5259 

7' 

610 

7S5330 

5401 

5472 

5543 

56i5 

5686 

5757 

5828 

5899 

5970 

7« 

611 

6041 

6112 

6i83 

6254 

6825 

6896 

6467 

6533 

6609 

6680 

71 

612 

6731 

6822 

6893 

6964 

7083 

7106 

7177 

724S 

7819 

7890 

7' 

6i3 

7460 

7531 

7602 

7678 

7744 

7815 

7885 

7956 

8027 

8098 

71 

614 

8168 

8289 

8810 

83  8 1 

8431 

8522 

8598 

8663 

8734 

8804 

71 

6i5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9869 

9440 

95 10 

71 

616 

9381 

9631 

9722 

9792 

9868 

9933 

»9»4 

"•74 

•144 

"215 

70 

617 

7902S5 

0356 

0426 

0496 

0367 

0687 

0707 

0778 

0S48 

0918 

70 

618 

0988 

1059 

1 1 29 

1 199 

1269 

1840 

1410 

1480 

1 530 

1620 

70 

619 

1691 

1761 

1881 

1901 

1971 

2041 

2111 

2181 

2252 

2822 

70 

620 

792892 

2462 

2532 

2602 

2672 

2742 

2812 

2SS2 

2952 

8022 

70 

621 

3092 

3i62 

323i 

33oi 

3871 

■3441 

35ii 

358i 

3631 

3721 

70 

622 

3790 

386o 

3980 

4000 

4070 

4189 

4209 

4279 

4849 

4418 

70 

623 

4488 

4558 

4627 

4697 

4767 

4886 

4906 

4076 

5045 

5ii5 

70 

624 

5i83 

5254 

5324 

5393 

5468 

5532 

56o2 

5672 

5741 

58ii 

70 

625 

588o 

5949 

6019 

6088 

6i58 

6227 

6297 

6366 

6486 

65o5  • 

69 

626 

6574 

6644 

6718 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

627 

7268 

7337 

7406 

7473 

7545 

7614 

7688 

7752 

7821 

7890 

69 

62S 

7960 

8029 

8098 

8167 

8286 

83o5 

8374 

8443 

83i3 

8582 

69 

629 

865 1 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

63o 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

69 

63 1 

800029 

0098 

0167 

02  36 

o3o5 

0878 

0442 

o5ii 

o58o 

0648 

69 

632 

0717 

0786 

0834 

0928 

0992 

1061 

1 1 29 

1 198 

1266 

i835 

69 

633 

1404 

1472 

i54i 

1609 

1678 

1747 

l8lD 

1884 

1952 

2021 

69 

634 

2089 

2i58 

2226 

2293 

2363 

2482 

25oo 

2568 

2637 

27q5 

635 

2774  2842 

2910 

2979 

3o47 

3ii6 

3 184 

3252 

3821 

3889 

6^ 

636 

3437 

8323 

3394 

3662 

3780 

3798 

3867 
4548 

3985 

4008 

4071 

68 

637 

638 

4189 

4208 

4276 

4344 

4412 

4480 

4616 

4685 

4753 

68 

4821 

4889 

4957 

5o23 

5og8 

5i6i 

5229 

3297 

5365 

5433 

68 

639 

55oi 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6ll3  1 

68 

N. 

0 

I 

a 

3 

4 

5 

6 

7 

8 

9 

D. 

A   TABLE  OP  LOGAEirnMS  FROM  1  TO  10,000. 


11 


"  N. 
640 

0 

I 

2 

3 

4 

5 

6 

- 

8 

9 

D. 

806180 

6248 

63i6 

6384 

6431 

6319 

6587 

6655 

6723 

6790 

68 

641 

6858 

6926 

6994 

706: 

7129 

7197 

7264 

7332 

7400 

7467 

68 

643 

7535 
621 1 

7603 

7670 
8346 

7738 

7806 

7813 
8549 

7941 

8008 

8076 

6143 

68 

643 

8279 

8414 

8481 

6616 

8684 

8751 

8818 

67 

644 

8886 

8933 

9021 

9088 

9136 

9223 

9290 

9358 

9425 

9492 

67 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

••3i 

••98 

•i65 

67. 

646 

810233 

o3oo 

0367 

0434 

o5oi 

o569 

o63& 

0703 

0770 

o837 

67 

647 

0904 

0971- 

1039 

1106 

1173 

1240 

i3o7 

1374 

1441 

i5o8 

67 

648 

1575 

1642 

1709 

1776 

1843 

191c 
"79 

1977 

2044 

2111 

2178 

67 

649 

2245 

23l2 

2379 

2445 

25l2 

2046 

2713 

2780 

2847 

67 

65o 

312913 

2980 

3047 

3ii4 

3i8i 

3247 

33i4 

3381 

3448 

35i4 

67 

65i 

358 1 

3648 

3714 

3781 

3848 

3914 

3o8i 

4048 

4114 

4181 

67 

65  2 

4248 

43i4 

438 1 

4447 

45i4 

4381 

4647 

4714 

4780 

4847 

67 

653 

49'3 

4980 

5o46 

5ii3 

5179 

5246 

53,12 

5378 

5445 

55ii 

66 

554 

5378 

5644 

3711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

655 

6241 

63o8 

6374 

6440 

65o6 

6373 

6639 

6705 

6771 

6838 

66 

656 

6904 

6970 

7o36 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

7365 
S226 

763 1 
8292 

7698 

7764 

7830 

7S96 
3556 

7962 

8028 

8094 

8i6o 

66 

658 

8338 

8424 

8490 

8622 

8688 

8754 

8820 

66 

659 

8885 

8931 

9017 

90t>3 

9149 

92i5 

9281 

9346 

9412 

9478 

66 

660 

819544 

9610 

0676 

9741 

9807 

9873 

9939 

•'•4 

••70 

•i36 

66 

661 

820201 

0267 

6333 

0399 

0464 

o33o 

0393 

0661 

0727 

0792 

66 

662 

o858 

0924 

0989 

1033 

1120 

1186 

1231 

i3i7 

i382 

1448 

66 

663 

i5i4 

1379 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2iq3 

65 

664 

2168 

2233 

2199 

2364 

243a 

2495 

2360 

2626 

2691 

2736 

65 

S65 

2822 

2887 

2932 

3oi8 

3o83 

3148 

32i3 

3279 

3344 

3409 

65 

b66 

3474 

3539 

36o5 

3670 

3735 

38oo 

3865 

3930 

3996 

4061 

65 

b68 

4126 

419: 

4236 

4321 

4386 

445 1 

45i6 

4381 

4646 

47 1 1 

65 

4776 

4841 

4906 

4971 

5o36 

5ioi 

5i66 

523i 

5296 

536i 

65 

669 

5426 

5491 

5556 

5621 

56Sb 

5751 

58i5 

588o 

5945 

6010 

65 

670 

826075 

6140 

6204 

6269 

6334 

6399 

6464 

6328 

6593 

6658 

65 

br- 

6723 

6787 

6832 

6oi7 

60S1 

7046 

7111 

7175 

7240 

73o5 

65 

572 

■7369 
!5oi3 

7434 

7499 

7363 

7628 

7692 

7757 
8402 

7821 
8467 

7886 
853 1 

7951 

65 

673 

80S0 

8144 

8209 

8273 

8338 

8395 

64 

Hi 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

675 

9304 

9368 

9432 

9497 

956 1 

9625 

9690 

9754 

9818 

9882 

64 

676 

99-17 

••11 

••75 

•139 

•204 

•26S 

•332 

•396 

•460 

'»525 

64 

678 

83o5rf9 

o653 

0717 

0781 

0845 

0909 

0973 

1037 

1678 

1102 

1 166 

64 

i23o 

1294 

i358 

1422 

i486 

i35o 

1C14 

1742 

1806 

64 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

238i 

2445 

64 

680 

8325o9 

2373 

2637 

2700 

2764 

2828 

2892 

2950 

3o20 

3o83 

64 

681 

3i47 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

682 

3784 

3tf48 

3912 

3975 

4039 

4io3 

4166 

423o 

4294 

4357 

64 

683 

4421 

4484 

4548 

4611 

4673 

4739 

4802 

4866 

4929 

4993 

64 

684 

5o56 

5 120 

5i83 

5247 

53io 

5373 

5437 

5300 

5564 

5627 

63 

685 

5691 

5754 

58i7 

588 1 

5944 

6007 

6071 

61 34 

6197 

6261 

63 

686 

6324 

63t;7 

6431 

65i4 

6377 

6641 

6704 

6767 

683o 

6894 

63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7523 

63 

688 

7388 
8219 

7632 

77i5 

8408 

7841 

7904 

7967 

So3o 

8093 

8i56 

63 

689 

8282 

8345 

8471 

8334 

8397 

8660 

8723 

8786 

63 

690 

838849 

8912 

S975 

9038 

9101 

9164 

92?7 

■9289 

935J2 

941 5 

53 

69  c 

9478 

9541 

9004 

9667 

9729 

9792 

9833 

0918 

99S1 

••43 

63 

(392 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

6545 

0608 

0671 

63 

693 

0733 

0796 

0839 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

694 

1359 

1422 

1483 

1347 

1610 

1672 

1735 

'797 

i860 

1922 

63 

695 

1083 

2047 

2110 

2172 

2235 

2297 

236o 

242-/ 

2484 

2547 

62 

696 

2609 

2672 

2734 

2796 

2839 

2921 

2983 

3046 

3ioS 

3170 

62 

697 

3233 

3295 

3357  . 

3420 

3482 

3544 

36o6 

3669 

3731 

3-93 

62 

698 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4413 

61 

699 

4477 

4539 

4601 

46t4 

4726 

4788 

485o 

4912 

4974 

5o36 

6a 

57 

N. 

0 

I 

3 

3 

4 

5 

6 

7 

8 

9 

1 

12 


A  TABLE   OF   LOGARITHMS   FROM   1   TO   10,000 


N. 

0 

I 

2 

3 

4 

5 

1  6 

7 

8 

9 

D. 

700 

843098 

5i6o 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

i3 

701 

5718 

5780 

5842 

5904 

5966 
6585 

6028 

6090 

6i5i 

6213 

6275 

t2 

702 

6337 

6399 

6461 

6523 

6646 

6708 

6770 

6832 

6894 

61 

7c3 

6g55 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

75ii 

62 

704 

7573 

7634 

7696 

S3 12 

7758 

7819 

So' 
8497 

8559 

8004 

8066 

8128 

62 

705 

8i8y 

825i 

8374 

8433 

8620 

8682 

87  0 

6: 

706 

88o5 

8866 

8928 

8989 

9031 

9112 

9174 

9235 

9297 

9358 

61 

707 

9"9 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

8joo33 

0095 

01 56 

0217 

0279 

o34o 

0401 

0462 

0324 

o585 

61 

709 

0646 

0707 

0769 

o83o 

0891 

0932 

1014 

1075 

ii36 

1197 

5i 

710 

851258 

l320 

i38i 

1442 

i5o3 

1 564 

1625 

1686 

1747 

1809 

61 

711 

1870 

io3. 

2541 

1992 

20  53 

2114 

2175 

2236 

2297 

2358 

2419 

6i 

712 

2480 

2602 

2663 

2734 

2785 

2846 

2907 

2968 

3029 

61 

713 

3090 

3i5o 

3211 

3272 

3333 

3394 

3455 

33i6 

3377 

3637 

61 

714 

3698 

3759 

3820 

388 1 

3941 

4002 

4o63 

4124 

4i85 

4245 

61 

715 

43o6 

4367 

4428 

4488 

4549 

4610 

4670 

473i 

4792 

4852 

61 

716 

4913 

4974 

5o34 

5095 

5i56 

5216 

5277 

5337 

5398 

5459 

61 

717 

5319 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6oo3 

6064 

61 

718 

6124 

6i85 

6245 

63o6 

6366 

6427 

6487 

6348 

6608 

6668 

60 

719 

6729 

6789 

685o 

6910 

6970 

7o3i 

7091 

7152 

7212 

7272 

60 

720 

357332 

7393 

7453 

75i3 

7574 

7634 

7694 

7755 

7815 
B417 

7875 

60 

721 

7935 

7995 

8o56 

8116 

8176 

8236 

8297 

8357 

8477 

60 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

9i38 

9198 

92  58 

9318 

9370 

9439 

9499 

9339 

9619 

9679 

60 

724 

9739 

9799 

9839 

99,8 
o5i8 

9978 

••38 

••98 

•i58 

•218 

•27S 

60 

725 

86o338 

0398 

0458 

0378 

0637 

0697 

0757 

0817 

0877 

60 

726 

0937 
1 534 

OQ96 

io56 

1116 

1 1 76 

1236 

1295 

i355 

i4i5   1475 

60 

727 
728 

1594 

1634 

1714 

1773 

i833 

1893 

1932 

2012 

2072 

60 

2l3l 

2191 

2231 

23io 

2370 

243o 

2489 

2349 

2608 

2668 

60 

729 

2728 

2787 

2S47 

2906 

2966 

3o25 

3o83 

3i44 

3204 

3263 

60 

730 

863323 

3382 

3442 

35oi 

356i 

3620 

368o 

3739 

3i99 

3858 

59 

73 1 

3917 

3977 

4o36 

4096 

4i55 

4214 

4274 

4333 

4392 

4452 

59 

732 

4311 

4370 

463o 

46S9 

4748 

4S08 

4867 

4926 

4985 

5043 

59 

733 

5io4 

5i63 

5222 

5282 

5341 

5400 

5459 

5319 

5578 

5637 

59 

734 

5696 

5755 

58i4 

5874 

5933 

5992 

6o5i 

6110 

6169 

6228 

59 

735 

6287 
6878 

6346 

6405 

6465 

6324 

6383 

6642 

6701 

6760 

6819 

59 

736 

6937 

6996 

7o55 

7114 

7173 

7232 

7291 

7350 

7409 

^9 

?S 

7467 

7526 

7385 

7644 

7703 

7162 

7821 
8409 

7880 

7939 

7998 

^9 

> 

8o56 

8ii5 

8174 

8233 

8292 

835o 

8468 

8327 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9o56 

9114 

9173 

59 

740 

869232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

^ 

741 

9818 

9877 

9933 

9994 

••53 

•ill 

•170 

•228 

•2S7 

•345 

5qi 

742 

870404 

0462 

0321 

0379 

o638 

0696 

0755 

o8i3 

0872 

0930 

743 

0989 
1573 

1047 

1106 

1 164 

1223 

1281 

i339 

1398 

1456 

I3l5 

58 

744 

i63i 

1690 

1748 

1806 

1 865 

1923 

1981 

2040 

2098 

58 

745 

2i56 

22l5 

2273 

233i 

2389 

2448 

25o6 

2564 

2622 

2681 

58 

746 

2739 

2797 

2855 

2913 

2972 

3o3o 

3o88 

3i46 

3204 

3262 

58 

74^ 

3321 

3379 

3437 

3495 

3353 

36ii 

3669 

2727 

3783 

3844 

58 

3902 

3960 

4340 

4oi8 

4076 

4i34 

4192 

4250 

43o8 

4366  4424 

58 

749 

4482 

4598 

4656 

4714 

4772 

■483  c 

4888 

4945 

5oo3 

58 

750 

875061 

5ii9 

5i77 

5235 

5293 

535i 

5409 

5466 

5524 

5582 

58 

75: 

5640 

5698 

5756 

58i3 

5871 

5929 

5987 

6045 

6102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

65o7 

6564 

6622 

6680 

6737 

58 

753 

6793 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

73i4 

58 

754 

7371 

7429 

7487 

7344 

7602 

7659 
8234 

77'7 

7774 

7832 

7889 

58 

755 

7Q47 

8522 

8004 

8062 

8119 

8177 

8292 

8349 

8407 

8464 

^' 

756 

8579 

8637 

8694 

8752 

8809 

8866 

S924 

898. 

9039 

57 

'  757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9335  1  961? 

^'^ 

i58 

9669 

9726 

9784 

9841 

9898 

9956 

••i3  ■ 

••70 

•127  1  •i85 

^ 

759 

880242 

0299 

o356 

o4i3 

0471 

0328 

o585 

0642 

0699  j  0756 

57 

I\. 

0 

I 

2 

3 

4 

5 

6 

7   ' 

8 

9 

D. 

A   TABLE   OF   LOGARITHMS   FROM   1    TO   10,000. 


13 


N. 
760 

0 

I 

I 

2 

3 

4 

5 

6 

7 

''  3 

9 

D. 

880814 

0871 

0928 

0985 

1042 

1099 

ii56 

I2l3 

1 27 1 

i328 

57 

76. 

i385 

1442 

U99 

1 556 

i6i3 

167c 

1727 

1784 

1841 

1898 

57 

762 

1955 

2D25 

2012 

2069 

2126 

2i83 

2240 

2297 

2354 

2411 

2468 

57 

763 

258i 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3o37 

5- 

764 

309J 

3i5o 

3207 

3264 

3321 

3377 

3434 

3491 
4059 
4623 

3348 

36o5 

St 

765 

366 1 

3718 

3775 

3832 

3888 

3945 

4002 

4ii5 

4172 

^7 

766 

4229 

4285 

4342  !  4399 

4455 

4312 

4569 

4682 

4739 
53o5 

57 

767 
768 

47Q3 

4852 

4909  1  4965 

5o22 

5078 

5i33 

5192 

5248 

57 

536 1 

5418 

5474 

533 1 

5587 

5644 

5700 

5757 

58i3 

5870 

u 

769 

5926 

5983 

6039 

6096 

6i52 

6209 

6265 

6321 

6378 

6434 

7:^ 

S86491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7o:>4 

7111 

7167  7223 

7280 

7336 

7392 
7955 

7449 

73o5 

7361 

56 

772 

7617 

7674 

7730  77S6 

7842 

7898 

8011 

8067 

8123 

56 

773 

8i7'y 

8236 

8292   8348 

8404 

8460 

83i6 

8573 

8629 

8685 

56 

774 

8741 

lltl 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

775 

9302 

9414 

9470 

9526 

9582 

9638 

9694 

9730 

9806 

56 

770 

9?62 

9918 

9974 

••3o 

••86 

•141 

•197 

•233 

*'^?2 

•365 

56 

777 

OQ0421 

0477 

o533 

o589 

0645 

0700 

0736 

0812 

0868 

0924 

56 

778 

0980 
1537 

io35 

logi 

1 147 

i2o3 

1259 

i3i4 

1370 

1426 

1482 

56 

779 

1393 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

893095 
at-Di 

2i5o 

2206 

2262 

23i7 

2373 

2429 

2484 

2540 

2595 

56 

78. 

2707 

2762 

2818 

2873 

2929 

2983 

3o4o 

3096 

3i5i 

56 

782 

<207 

3262 

33i8 

3373 

3429 

3484 

3340 

3595 

365i 

3706 

56 

783 

'£1 

3817 

3873 

3928 

3984 

4039 

4094 

4130 

42o5 

4261 

55 

784 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

49S0 

5o36 

5091 

5146 

3201 

5237 

53i2 

5367 

55 

786 

5423 

5473 

5333 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

787 

5975 

•6326 

6o3o 

6oS5 

6140 

6195 

625: 

63o6 

636i 

6416 

6471 

55 

788 

658 1 

6636 

6692 

6747 

6S02 

6857 

6912 

6967 

7022 

55 

789 

7077 

7i32 

7187 

7242 

7297 

7352 

7407 

7462 

7317 

7572 

55 

790 

397627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

79' 

8176 

823i 

8286 

8341 

8396 

8431 

8306 

856 1 

86i5 

8670 

55 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

93  S3 

9437 

9492 

9547 

9602 

9636 

971 1 

9766 

55 

794 

9821 

9875 

9930 

99S3 

••39 

••94 

*i49 

•203 

•258 

•3l2 

55 

793 

900367 

0422 

0476 

033 1 

o586 

0640 

0695 

0749 

0804 

0859 

55 

796 

ogi3 

096a 

1022 

1077 

I  i3i 

1186 

1240 

1293 

1349 

1404 

55 

797 

1458 

i5i3 

1 567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

798 

2oo3 

2057 

21(2 

2166 

2221 

2275 

2329 

2384 

2438 

2402 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3o36 

54 

800 

903090 

3i44 

3199 

3233 

3307 

3361 

3416 

3470 

3524 

357S 

54 

801 

3633 

3687 

3741 

3795 

3S49 

3904 

3958 

4012 

4066 

4120 

54; 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 
5i4S 

'4661 

54  j 

8o3 

4716 

4770 

4824 

4878 

4932 

4986 

3o4o 

5094 

5202 

541 

B04 

5256 

53io 

5364 

5418 

5472 

5J26 

558o 

5634 

5688 

5742 

54 

boD 

5796 

585o 

5904 

5958 

6012 

6066 

61 19 

6173 

6227 

6281 

54 

S06 

6335 

()389 

6443 

6497 

655 1 

66o4 

6658 

6712 

6766 

6820 

54 

837 
808 

6874 

6927 

6981 

7033 

70S9 

7143 

7196 

725o 

73o4 

733S 

54 

7411 

7465 

7319 

7573 
8110 

7626 

768c 

7734 

7787 

7841 
3378 

7895 

54: 

809 

7949 

8002 

8o56 

8i63 

8217 

6270 

8324 

8431 

54  j 

81C 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

54! 

Bii 

9021' 

9074 

9128 

9181 

9235 

92S9 

9342 

93q6 

9449 

93o3 

54 

812 

9556 

9610 

9663 

9716 

977f^ 

9823 

0877 

9930 

9984 

••37 

53 

8i3 

9 1 0C9 1 

J144 

0197 

o25i 

o3o4 

o358 

041 1 

0464 

o5i8 

057. 

53 

814 

0624 

0678 

073 1 

0784 

o338 

0891 

0944 

0998 

io5i 

1104 

53 

Bid 

ii58 

1211 

1264 

i3i7 

1371 

1424 

1477 

i33o 

1 584 

1637 

53 

816 

1690 

1743 

1797 

i85o 

1903 

1936 

2009 

2o63 

2116 

2169 

53 

817 

2222 

2273 

2328 

238i 

2435 

2488 

2541 

2594 

2647 

2700 

53 

818 

2753 

2806 

2859  2913 

2966 

3019 

3072 

3l23 

3178 

323i 

53 

819 

3284 

3337 

3390  3443 

3496 

3549 

36o2 
6 

3655 

1 

3708 

3761 

53 

N. 

0 

I 

a 

3 

4 

5 

7 

8 

1 
9 

D. 

u 

A 

TABLE  OF  LOOAltlTUMS  FKO.M 

1  TO 

10.000. 

N. 

0 

'      i  ' 

3 

4   1   3 

6 

7 

8 

0 

D. 

53 

820 

913S14!  3867  1  3920 

3973 

4026 

!  4079 

4i32 

4184 

4237 

4290 

83  i 

4343;  43p6 

4449 

4302 

4555 

i  4608 

4660 

4713 

4766 

4819 

53 

822 

4872,  4925 

4977 

5o3o 

5o83 

5i36 

5189 

5241 

5294 

5347 

53 

823 

5400!  5453 

55o5 

5558 

56 II 

5664 

5716 

5769 

5822 

3875 

53 

824 

5927'  5980 

6o33 

6o85 

6i38 

6191 

6243 

6296 

6349 

6401 

C? 

8i5 

64J4|  6007 

6309 

6612 

6664 

1  &717 

6770 

6822 

6875 

6927 

53 

826 

1  6980  7033 

70S5 

7i38 

7190 

7243 

7295 

7348 

7400  1  7453 

53. 

^'« 

7306  7558 

7011 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

5j 

82!3 

1  8o3c  8o83 

8i35 

8188 

8240 

8293 

8345 

8397 

8450 

85o2 

52 

829 

8553  8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

83o 

919078 

9i3o 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

83 1 

9601 

9653 

9706 

9738 

9810 

9862 

99 '4 

9967 

••19 

••71 

52! 

832 

9201231  0176 

0228 

0280 

o332 

0384 

0436 

0489 

o54i 

0393 

52 

833 

0645 

0697 

0749 

080 1 

o853 

0906 

0958 

1010 

1062 

II 14 

52 

834 

1166 

1218 

1270 

l322 

1 374 

1426 

1478 

i53o 

i582 

i634 

52 

835 

1686 

1738 

1790 

1842 

1894 

19/(6 

1998 

2o5o 

2102 

2 1 54 

5^ 

836 

2206 

2258 

23lO 

2362 

2414 

2466 

25i8 

2370 

2622 

2674 

52 

837 

725 

2777 

2S29 

2881 

2933 

2985 

3o37 

3089 

3 1 40 

3192 

52 

838 

3244 

3296 

3348 

3399 

.345 1 

33o3 

3555 

3607 

3658 

3710 

52 

839 

37OL) 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

32 

840 

924279 

433 1 

4383 

4434 

44S6 

4538 

4389 

4641 

4693 

4744 

52 

841 

4790 

4848 

4899 

4931 

5oo3 

5o54 

5 106 

5i57 

5209 

5261 

52  1 

842 

53 1 2 

5364 

541D 

5467 

5318 

5570 

5621 

5673 

5723 

5776 

52 

843 

5828 

5879 

5931 

5982 

6o34 

6o85 

6137 

6188 

6240 

6291 

5i 

844 

6342 

6394 

6445 

6497 

6348 

6600 

665 1 

6702 

6754 

68o5 

'n 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

JI 

846 

7370 

7422 

7473 

7324 

7576 

7627 

7678 

7730 

7781 

7832 

5i 

847 

7883 

7935 

7986 

8037 

8088 

8140 

9191 

8242 

8293 

8345 

5i 

848 

8396 

B447 

8498 

8549 

8601 

8652 

8703 

8754 

88o5  8837 

5i 

849 

8908 

8959 

9010 

9061 

9112 

9i63 

9215 

9266 

9317 

9368 

5i 

85o 

929419 

9470 

9521 

9572 

9623 

9674 

9723 

9776 

9827 

9879 

5i 

85 1 

9930  Qg8l 

••32 

••83 

•j34 

•i85 

•236 

•287 

•338 

•389 

5i 

852 

930440 

0491 

o542 

0592 

0643 

0694 

0743 

0796 

0847 

0898 

5i 

853 

0949 

1000 

io5i 

1102 

II33 

1204 

1234 

i3o5 

i356 

1407 

5i 

854 

1458 

i3o9 

i56o 

1610 

106 1 

1712 

1763 

1814 

i865 

1915 

5: 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

5i 

B56 

2474 

2524 

2375 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

5i 

857 

2981 

3o3i 

3  08  2 

3i33 

3i83 

3234 

3285 

3335 

3386 

3437 

5i 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

5i 

859 

3993 

4044  4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

5i 

860 

934498 

4549 

4599 

4650 

4700 

4751 

4'!oi 

4832 

4902 

4953 

5o 

861 

5oo3 

5o54 

5 104 

5i54 

52o5 

5255 

53o6 

5356 

5406 

5457 

5o 

8C12 

5.07 

5558 

56o8 

5658 

5709 

5759 

5809 

5860 

3910 

5960 

5o 

8t3 

601 1 

6061 

6111 

6162 

6212 

6262 

63i3 

6363 

6413 

6463 

5o 

864 

65i4 

6564 

6614 

6665 

6715 

6765 

681 5 

6865 

6916 

6066 

5o 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

5o 

866 

7318J  7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

5o 

867 

8019  8069 

81 19  1  8169  j 

8219 

8260 

8320 

8370 

8420 

8470 

5o 

868 

8520  8570 

8620 

8670 

8720 

8770 

8«20 

8870 

8920 

8970 

5o 

869 

9020  9070 

9120 

91-0 

9220 

9270 

9320 

9369 

9419 

9469 

5o 

870 

939619  9569 

9619 

9069 

9719 

9769 

9819 

9869 

9918 

9968 

5o 

^7' 

940018  0068 

0118 

0168 

0218 

0267 

o3i7 

o367 

0417 

0467 

5o 

872 

o5i6  o566 

0616 

0666 

07 1 6 

0765 

08 1 5 

o865 

09 1 5   0964 

5o 

873 

1014'  1064 

1114 

ii63 

I2l3 

1263 

i3i3 

1 362 

1412   1462 

5o 

874 

i5ii|  i56i 

1611 

1660 

1710 

1760 

1809 

1839 

1909   1958 

5o 

875 

2008  2o58 

2107 

2157 

2207 

2236 

23o6 

2355 

24o5 

2433 

5o 

876 

25o4 

2554 

2603 

2653 

2702 

2752 

2801 

2S5i 

2901 
3396 

2960 

5o 

877 

3ooo 

3  049 

3099 

314B 

3198 

3247 

3297 

3346 

3445 

49 

878 

3495(  3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

393q 

49 

879 

3989'  4o38 

4088 

4137 

4186 

4236 

4285 

4335 

4384  4433  1 

49 

0 

I 

2    3 

4 

5 

6 

7     8 

9 

"dT 

A  TABLE 

OF  LOGARITUMS  FROM  1 

TO  10,000 

. 

16 

IT. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

8So 

9444<S3 

4532 

458 1 

463 1 

4680 

4729 

4779 

4828 

4877 

4017 

49 

881 

4976:  So25 

5074 

5i24 

5.73 

3222 

5272 

5321 

5370 

5419 

49 

8S2 

5.I69  55 18 

5567 

56i6 

5665 

5715 

5764 

58i3 

5862 

591  2 

49 

883 

5o6i  6oio 

6o5q 

6108 

6157 

6207 

6256 

63o5 

6354 

6405 

49 

884 

6j5j  65oi 

655 1 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

^9 

885 

6943;  6Qg2 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7383 

49 

886 

7434 

7483 

7532 
6022 

75Si 

7630 

7679 

7728 

7777 

7«26 

7875 

49 

887 

7924  7073 

8070 

8119 

8168 

82.7 

8266 

83i5 

8364 

49 

888 

8413 

8462 

85n 

836o 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 

949390 

9439 

9488 

9536 

9585 

9634 

9683 

973 1 

9780 

9829 

49 

891 

9^76 

9926 

9975 

••24 

••73 

•121 

•170 

•219 

•267 

•3 16 

49 

892 

930365 

0414 

0462 

o5ii 

o56o 

0608 

0657 

0706 

0734 

oSo3 

49 

893 

oBoi 

0900 

0949 

0997 

1046 

1095 

1 143 

1192 

1240 

1289 

49 

894 

1 338 

i386 

1435 

1483 

i532 

i5oo 

1629 

1677 

1726 

1773 

H 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

896 

23ob 

2356 

24o5 

2433 

25o2 

2530 

2399 

2647 

2096 

V^i 

48 

897 

2792 

2841 

2889 

2938 

29S6 

3o34 

3o83 

3i3i 

3i>-'o 

3228 

48 

898 

3276 

3323 

3373 

3421 

3470 

3318 

3566 

36i5 

3663 

3711 

48 

S99 

3760 

38o8 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

9342*3 

4291 

4339 

4387 

4435 

4484 

4532 

4380 

4628 

4677 

48 

901 

4725 

4773 

4821 

4860 

4918 

4966 

5o!4 

5o62 

5iio 

5i58 

48 

902 

3207 

5235 

53o3 

53:, 

5399 

588o 

5447 

3493 

5543 

5392 

5640 

48 

903 

50S8 

5735 

57K'. 

58  J  2 

5928 

5976 

6024 

6072 

6120 

48 

904 
905 

6168 

6216 

6263 

63 1 3 

636 1 

6409 

6457 

65o3 

6353 

6601 

48 

6049 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7o32 

7080 

48 

906 

7126 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7312 

7539 

48 

907 

7007 

7655 

7703 

7731 

7799 

7847 

7«94 

7942 

P'>Z 

8o38 

48 

908 

80S6 

8i34 

81S1 

8229 

8277 

8325 

8373 

8421 

8408 

85i6 

48 

909 

8564 

86i2 

8639 

8707 

8755 

88o3 

835o 

8898 

8946 

8994 

48 

910 

939041 

9089 

9137 

91S5 

9232 

9280 

o328 

9375 

9423 

9471 

48 

911 
912 

9318 
9995 

9366 
•042 

9614 
••90 

9661 
01 38 

9709 

•l83 

9757 
0233 

9804 
•280 

o852 
•328 

9900 
•376 

9947 
•423 

48 
48 

913 

960471 

o5i8 

0366 

06 1 3 

0661 

0709 

0736 

0804 

o83i 

0S99 

48 

914 

0946 

0994 

1041 

1089 

ii36 

1 1 84 

I  23  I 

1279 

i326 

1374 

47 

910 

1421 

1409 

I3l6 

1363 

1611 

i658 

1706 

1733 

iSoi 

1848 

47 

916 

1S95 

I9i3 

1990 

2o38 

2o85 

2l32 

2180 

2227 

2275 

2322 

47 

9'7 

2369 

2417 

2464 

25ll 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3o32 

3079 

3i26 

3174 

3221 

3268 

47 

919 

33i6 

3363 

3410 

3457 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4i65 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4493 

4542 

4390 

4637 

4684 

47 

922 

4731 

4773 

4825 

4872 

4919 

4966 

5oi3 

5o6i 

5io8 

5i55 

47 

923 

3202 

3249 

5296 

5343 

5390 

5437 

5484 

553 1 

5578  ■ 

5625 

47 

924 

0672 

57.9 

5766 

58i3 

5S6o 

5907 
6376 

5934 

6001 

6048 

6095 

47 

925 

6142 

5 1 8. 

6236 

6283 

6329 

6423 

6470 

6317 

6364 

41 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127 

7.73 

7220 

7267 

73 1 4 

736i 

7408 

7454 

73oi 

47 

928 

7548 

7393 

7642 
8109 

76H8 

7735 

7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

Si56 

8203 

8249 

8296 

8343 

8390 

8436 

4'; 

93d 

968483!  8530 

8576 

8623 

8670 

S716 

8763 

8810 

8856 

8903 

47 

^, 

8930  8996 

9043 

9090 

91 36 

9i83 

9229 

9276 

9323 

9369 

47 

Vl2 

0416I  9463 

95og 

9356 

9602 

9649 

9695 

9742 

97S9 

9S33 

47 

933 

98821  9928 

9973 

••21 

••68 

•ii4 

•161 

•207 

•234 

•3oc 

47 

934 

970347!  0393 

0440 

0486 

o533 

0379 

0626 

0&72 

°'^IP, 

0763 

46 

933 

38121  o853 

0904 

0931 

0997 

1044 

1090 

1 137 

ii83 

12:9 

46 

936 

1276!  i322 

1369 

1413 

1461 

i5o3 

1 534 

1601 

1647 

1693 

46 

937 

1740;  1786 

i832 

1S79 

1925 

1971 

2018 

2064 

2110 

2137 

46 

938 

2203  2249 

2295 

2342 

2383 

2434 

2481 

2327 

2373 

2619 

46 

939 

2666'  2712 

2738 

2804 

285i 

2897 

2943 

2989 

3o35 

3o82 

46 

1  ■ ~ 

N. 

1 
0   1   I 

^ 

3 

4 

5 

6 

7 

8 

9 

D. 

26 


16 


A   TABLE    OF   LOGARITUMS    FROM    1    TO    10,000. 


r-  "  ■ 

N. 

0 

1 

I   1  1 

3 

1  4 

5 

6     7 

8 

9 

1  ]). 

1 

940 

973128  3174 

3220 

3266 

33i3 

3359 

34o5 

345i   3497 

3543 

46 

941 

3590  3036 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4oo5 

46 

942 

4oDi  4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

943 

4312!  4338 

4604 

465o 

4696 

4742 

47S8 

4834 

4880 

4926 

46 

944 

4972  5oiH 

5064 

5iio 

5i36 

5202 

5248 

5294 

5340 

5386 

46 

945 

54J2  5478 

5324 

5570 

56 16 

5662 

5707  1  5733  1  5799 

5845 

4C 

y46 

5891,  5937 

3983 

602Q 

6o75 

6121 

6167  ,  6212 

6238 

63o4 

46 

947 

6330  63j6 

6442 

6488 

6533 

6579 

6625  1  6671 

6717 

6763 

46 

948 

680S  6834 

6900 

7358 

6946 

6992 

7037 

7083 

7129 

7173 

7320 

:  4<' 

i 

949 

7266  7312 

74o3 

7449 

7495 

7541 

7586 

7632 

7678 

4t> 

1 

i 

9J0 

977724'  7769 

7815 

7861 

7906 
8363 

7952 

7998 

8043 

8089 

8i35 

i  46 

95 1 

8181:  8226 

8272 

83i7 

8409 

8434 

85oo 

8546 

8391 

46 

962 

8637  8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

9093  91 38 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

93o3 

46 

954 

9548  9394 

9639 

9685 

9730 

9776  1  9821 

9867 

9912 

9953 

46 

955 

9S0003  0049 

oog4 

0140 

0185 

023l 

0276 

0322 

o367 

0412 

45 

936 

0438  o3o3 

0349 

0594 

0640 

o685 

0730 

0776 

0821 

0867 

45 

957 

09 1 2  0957 

ioo3 

1048 

1093 

1139 

1184 

1229 

1275 

l320 

45 

9.58 

1 360  1 4 1 1 

1456 

i5oi 

1547 

1592 

1637 

i683 

1728 

1773 

45 

959 

1819  1864 

1909 

1954 

2000 

2045 

2090 

2i35 

2181 

2?35 

45 

960 

982271  23i6 

2362 

2407 

2432 

2497 

2543 

2588 

2633 

267vi 

45 

961 

2723  2769 

2814 

2839 

2904 

3356 

2949 

2994 

3  040 

3o83 

3i3o 

45 

962 

3175  3220 

3265 

33io 

3401 

3446 

3491 

3536 

3581 

45 

963 

3626;  3671 

3716 

3762 

3807 

3S52 

3S97 

3942 
4392 

39S7 

4o32 

45 

964 

4077  4122 

4167 

4212 

4257 

43o2 

4347 

4437 

44S2 

45 

965 

4327'  4372 

4617 

4662 

4707 

4752 

4797 

4842 

48S7 

4q32 

45 

966 

4977  5o22 

5067 

5ll2 

5,37 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426;  5471 

55i6 

556 1 

56o6 

5631 

56g6 

5741 

5786 

5830 

45 

968 

5875!  5920 

5965 

6010 

6035 

6100 

6144 

6189 

6234 

6279 

45 

969 

6324,  6369 

64i3 

6458 

65o3 

6348 

6593 

6637 

66S2 

6727 

45 

970  986772'  6817 

6861 

6906 

6951 

6996 

7040 

7085 

7i3o 

7175 

45 

97' 

7219  7264 

7309 

7353 

7398 

7443 

74S8 

7532 

7577 

7622 

45 

972 

7666;  771 1 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

973 

8ii3  8137 

8202 

8247 

82QI 

8336 

838i 

8423 

8470 

85i4 

45 

974 

8559  8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9003  9049 

9094 

91 38 

9i83 

9227 

9272 

9316 

9361 

9403 

45 

976 

g45o'  9494 

9893,  9939 

9903391  o3h3 

9539 

9583 

9628 

9672 

97'7 

9761   9806 

9830 

44 

977 
978 

9983 

••28 

••72 

•117 

•161 

•206  :  "250 

•204 

44 

0428 

0472 

o5i6 

o56i 

o6o5 

o65o 

0694 

0738 

44 

979 

0783  0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

1 

980 

99:226  1270 

i3i5 

i339 

i4o3 

t448 

1492 

1 536 

i58o 

1625 

44 

981 

1669  1713 

1758 

1802 

1846 

l8go 

ig35 

"979 

2023 

2067 

44 

982 

2111!  2 1 56 

2200 

2244 

22S8 

2333 

2377 

2421 

2465 

2309 

44 

983 

2554!  2398  2642 
2995^  3o39  3o83 

2686 

2730 

2774 

28ig 

2863 

2907 

2931 

44 

1 

984 

3127 

3172 

32i6 

3260 

33o4 

3348 

33g2 

'4^ 

1 

985 

3436  3480  3524 

3568 

36 1 3 

3657 

3701 

3745  37S9 

3833 

44 

1 

986 

3STi'   3921   3965 

4009 

4o53 

4097 

4141 

4i85   4229 

4273 

44 

i 

987 

4317:  436i 

44o5 

4449 

4493 

4537 

458 1 

4625  1  4069  , 

47i3 

4.'. 

988 

4757'  48oi 

4845 

4889 

4g33 

4977 

502l 

5o65  1  5io8 

5i52 

44 

989 

5196;  5240 

5284 

5328 

5372 

5416 

5460 

55o4  5547 

559. 

44 

990 

995635  5679 

5723 

5767 

58ii 

5854 

5898 

5942   5986 

6o3o 

44 

M 

991 

6074'  61 17 

6161 

6203 

6249 

6293 

6337 

638o 

6424 

6468 

44 

■ 

992 

6312  6555 

65g9 

6643 

6687 

6731 

6774 

6818 

6862 

6go6 

44 

T 

993 

6949  6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 
7736 

7343 

44 

994 

7386'  743o 

7474 

7517  ; 

7561 

7605 

764S 

7692 

7779 

44 

993 

7823  7867 

7910 

7954 

7998 

8041 

8o85 

8129 

8172 

8216 

44 

996 

8239 

83o3 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

m 

997  1  8693 

8739 

8782 

8826 

8869 

8gi3 

8936 

9000 

9043 

9087 

44 

I 

998   9i3i 

9174 

9218 

9261 

93o5 

934S 

9392 

9435 

9479 

9522 

44 

■ 

999  1  9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

99'3 

9957 

43 
D. 

1 

N.  i  0     I 

1 

2 

3 

4 

5 

6 

7 

8 

9 

A  TABLE 


OP 


LOGAEITHMIC 


SINES   AND  TANGENTS 


FOR   EVEBT 


DEGREE  AND  MINUTE 
OF  THE  QUADRANT. 


Remark.  The  minutes  in  the  left-hand  column  of  eacli 
page,  Increasing  downwards,  belong  to  the  degrees  at  the 
top ;  and  those  increasing  upwards,  in  the  right-hand  coiunin, 


belong  to  tho  degrees  below. 


18 


(0    DEGREES.)      A   TABLE    OF    LOGARITIIMIC 


0 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotaiig. 

0-000000 

10-000000 

0-000000 

Infinite. 

60 

I 

6-463726 

5017-17 

00000c 

-00 

6-463726 

5017- 

17 

13-536274 

59 

2 

764756 

2934- 

85 

000000 

•  00 

764756 

2934- 

83 

235244 

58 

3 

940S47 

2082- 

3i 

000000 

-00 

9io.S47 

2082- 

3i 

059153 

57 

4 

7-065786 

i6i5- 

6^ 

000000 

•  00 

7-065786 

i6i5- 

17 

12-934214 

56 

5 

162696 

i3i9- 

000000 

-00 

162690 

i3i9- 

60 

837304  :  VJ    1 

6 

241877 

iii5- 

75 

9-999999 

•  01 

241878 

iii5- 

78 

758122  1  54 

n 

308824 

066- 

53 

999999 

-01 

308825 

At 

53 

691175  '  5} 

g 

3668 1 6 

852- 

54 

999999 

-01 

366817 

54 

633 1 83  1  52 

9 

417968 

762- 

63 

999999 

•01 

417970  1 

762- 

63 

582o3o  ■  5i 

10 

463725 

689- 

83 

990998 

•01 

463727 

689- 

88 

536273   " 

3o 

II 

7-5o5ii8 

629- 

81 

9-999998 

•  01 

7 -505 1 20 

629- 

81 

12-494880 

49 

12 

542906 

579- 

36 

999997 

•  01 

542909 

^]?' 

33 

4D7091 

48 

i3 

577668 

536- 

41 

999997 

•  01 

577672 

536- 

42 

422328 

47 

14 

609853 

499- 

38 

999996 

-01 

609857 

499 

H 

390143 

46 

if 

639816 

467- 

14 

999996 

•  01 

639820 

467. 

i5 

36oi8o 

45 

i6 

667845 

438- 

81 

999995 

•  01 

667849 

438 

82 

332 i5i 

44 

17 

694173 

4i3- 

72 

999995 

-01 

694179 

4i3 

73 

3o582i 

43 

i8 

718997 

39 1- 

35 

999994 

-01 

719004 

391 

36 

2S0997 

42 

19 

742477 

371- 

27 

999993 

-01 

742484 

371 

28 

257.M6 

41 

20 

764754 

353- 

i5 

999993 

•  01 

764761 

35i 

36 

235239 

40 

21 

7-785943 
806146 

336- 

72 

9-999992 

•01 

7.785951 

336 

73 

12-214049 

39 

22 

321- 

75 

999991 

-01 

8061 55 

321 

76 

193843 

38 

23 

825451 

3o8- 

o5 

999990 

-01 

825460 

3o8 

06 

174540 

37 

24 

843934 

2n5 

U 

999989 

•02 

843944 

295 

49 

i56o56 

36 

25 

861662 

283 

999988 

-02 

861674 

283 

90 

i38326 

35 

26 

878695 

273 

17 

999988 

-02 

878708 

273 

18 

121292 

34 

27 
28 

893085 
910879 

263 
253 

23 

99 

999987 
999986 

■  02 

-02 

895099 
910894 
926134 

263 
254 

25 

01 

104901 
089106 

33 

32 

29 

926119 

245 

38 

999985 

-02 

245 

40 

0738&6 
039142 

3i 

3o 

940842 

237 

33 

999983 

-02 

940858 

237 

35 

3o 

3i 

7-9550S2 

229 

80 

9-999982 

-02 

7-955100 

229 

81 

1 2 • 044900 

29 

32 

Q68870 

222 

73 

999981 

-02 

968889 
982253 

222 

75 

o3iiii 

28 

33 

982233 

216 

08 

999980 

-02 

216 

!0 

017747 

27 

34 

993198 

20Q 

81 

999979 

-02 

993219 

20g 

83 

004781 

26 

35 

8-007787 

2o3 

90 

999977 

-02 

8-007809 

2o3 

02 

11-992191 

970955 
968055 

25 

36 

020021 

K;H 

3 1 

999976 

■02 

020045 

198 

33 

24 

37 

o3igi9 
043301 

iq3 

02 

999975 

-02 

o3 1 945 

193 

188 

o5 

23 

33 

188 

01 

999973 

-02 

043527 

o3 

956473 

22 

39 

054761 

1 83 

25 

999972 

-02 

054809 

1 83 

27 

943191 

21 

40 

065776 

I,!8 

72 

999971 

•02 

o658o6 

178 

74 

934194 

20 

41 

8-076500 

'74 

41 

9-999960 
999968 

-02 

8-076531 

174 

44 

11-923469 

IQ 

42 

086965 

170 

3 1 

-02 

086997 

170 

34 

9i3oo3 

18 

43 

097 1  S3 

166 

3q 

999966 

-02 

097217 

166 

42 

902783 
892797 
S83o37 

n 

44 

107167 

162 

65 

999964 

•o3 

107202 

162 

68 

16 

45 

1 16926 

159 

08 

999963 

•o3 

1 1 6963 

i5g 

10 

i5 

46 

1 2647 1 

i5d 

66 

999961 

•o3 

i265:o 

i55 

68 

873490   14 

47 

i358io 

l52 

38 

900959 

•o3 

1 3585 1 

l52 

41 

864149  1  i3 

48 

144953 

149 

24 

999968 

-o3 

144096 

149 

-27 

855oo4 

12 

49 

153907 

146 

22 

999,y5j 

•o3 

153952 

146 

•27 

846048 

u 

5o 

1626S1 

143 

33 

999954 

•o3 

162727 

143 

-36 

837273 

10 

5i 

8-171280 

140 

54 

9-999952 

•o3 

8-171328 

140 

•57 

11-828672 

I 

52 

179713 

i37 

86 

999950 

•o3 

179763 

i37 

.00 

820237 

53 

187985 

i35 

29 

999948 

•  03 

iB8o36 

i35 

-32 

811964 

1 

54 

196102 

l32 

80 

999946 

-o3 

196156 

l32 

84 

8o3844 

6 

55 

20.',o70 

i3o 

41 

999944 

-o3 

204126 

i3o 

-44 

795874 
788047 

5 

56 

211895 

128 

10 

999942 

•04 

211953 

128 

■14 

4 

57 

219581 

125 

-87 

999940 

•  04 

219641 

125 

•  90 

780359 

3 

58 

227134 

123 

72 

999938 

.04 

227195 

123 

-76 

772S03 

a 

59 

234557 
24i855 

121 

64 

999936 

•  04 

234621 

121 

-68 

765379 

I 

66 

119-63 

999934 

-04 

241921 

119-67 

75S079 

0 

Cosine 

D. 

Sine 

!  Cotang. 

D. 

Tang. 

M. 

(89    DEGREES.) 


SIXES  AND  TANGENTS.       (1    DEGREE.) 


19 


M. 

Sine 

D. 

Cosine  1 

D. 

Tang. 

D 

Cotivng. 

0 

8-;4i855 

119 

63 

9-999934 

-04 

8-24I921 

119 

t-/ 

11  758379 

60 

I 

249033 

117 

68 

999932 

•04 

249102 

117 

72 

75oS9§ 

^ 

i 

256094 

ii5 

So 

999929 

•  04 

256i65 

ii5 

64 

743835 

W 

3 

263042 

ii3 

98 

999927 

•04 

263ii5 

114 

02 

736885 

ll 

4 

269881 

112 

21 

999923 

•  04 

269956 

112 

25 

730044 

56 

(C 

276614 
28^43 

1 10 

5o 

999922 

•  04 

276691 

no 

54 

723309 

55 

I 

loS 

83 

999920 

•  04 

283323 

108 

87 

716677 

54 

I 

239773 

107 

21 

999918 

•  04 

289856 

107 

26 

710144 

53 

296207 

io5 

65 

999915 

•  04 

296292 

io5 

70 

703708 

52 

9 

302546 

104 

1 3 

999913 

•  04 

3o2634 

104 

18 

697366 

5i 

10 

308794 

102 

66 

999910 

•  04 

308884 

102 

70 

691116 

5o 

II 

8-3i49o4 

101 

22 

9-999907 

•  04 

8-3i5o46 

101 

26 

■1-684954 

49 

12 

321027 

99 

82 

999905 

•  04 

321122 

99 

87 

67S878 

48 

i3 

327016 

98 

47 

999902 

•ci 

327114 

98 

5i 

672886 

47 

14 

332924 

97 

14 

999899 

-05 

333025 

97 

19 

666975 

46 

i5 

338^53 

95 

86 

999S97 

•  o5 

338856 

95 

90 

661144 

45 

i6 

344504 

94 

60 

999894 

•o5 

344610 

94 

65 

655390 

44 

\l 

35r.i8i 

93 

38 

999801 
999888 

.o5 

350289 

93 

43 

6497 ' ' 

43 

355783 

92 

0^ 

•  o5 

3558o3 
36i4Jo 

92 

24 

644 io5 

42 

"9 

36i3i5 

91 

999885 

-o5 

9' 

08 

63B570 

41 

20 

366777 

89 

90 

999882 

.o5 

366895 

89 

95 

633 1 o5 

40 

21 

8-372171 

88 

80 

9-999879 

•  05 

8-372292 

88 

85 

11-627708 

39 

22 

377499 

87 

72 

999876 

-03 

377622 

87 

77 

622378 

38 

23 

382762 

86 

67 

999873 

•  o5 

382889 

86 

72 

617111 

37 

24 

387962 

85 

64 

999870 

•  o5 

388092 

85 

70 

611908 

36 

25 

393101 

84 

64 

999867 

-05 

393234 

84 

70 

606766 

35 

26 

398179 

83 

66 

999864 

.o5 

398315 

83 

71 

601685 

34 

3 

4o3i99 

82 

71 

999861 

•  o5 

403338 

82 

76 

596662 

33 

408161 

81 

77 

999808 

•  05 

4o83o4 

81 

82 

09 1 696 

32 

=9 

4i3o68 

80 

86 

9998S4 

•  05 

4i32i3 

80 

91 

586787 

3i 

3o 

417919 

79 

96 

999801 

.06 

418068 

80 

02 

581932 

3o 

3i 

8-422717 

79 

09 

23 

9  •  9998/18 

.06 

8-422869 

79 

14 

ii-577i3i 

29 

32 

427462 

7S 

999844 

.06 

427618 

78 

3o 

572382 

28 

33 

432156 

77 

40 

999841 

-06 

43231 5 

77 

45 

567685 

27 

34 

436800 

76 

57 

99983B 

•  06 

436062 

76 

63 

563o38 

26 

35 

441394 

75 

77 

999834 

.06 

44i36o 

75 

83 

558440 

25 

36 

445941 

74 

99 

99983  i 

.06 

446 1 1 0 

75 

o5 

553SOO 

24 

\l 

450440 

74 

22 

999827 
99982J 

-d6 

45o6i3 

74 

28 

549387 

23 

454893 

73 

46 

•06 

455070 

73 

52 

544930 

22 

39 

459301 

72 

73 

999S20 

-06 

459481 

72 

79 

54o5i9 

21 

40 

463665 

72 

00 

999816 

-06 

463849 

72 

06 

536i5i 

20 

41 

8-467985 

71 

29 

9-999812 

■  06 

&-468172 

71 

35 

11-531828 

19 

42 

4-2263 

70 

60 

999809 

•  06 

472454 

70 

66 

527346   18 

43 

476498 

69 

91 

999805 

-06 

476693 
480892 

69 

f 

523307 

17 

44 

480693 

69 

24 

999801 

-06 

69 

3i 

519108 

16 

45 

484848 

68 

59 

009797 

-07 

485o3o 

68 

65 

5i4q5o 

i5 

4& 

488963 

67 

t 

vy;793 

•  07 

489170 
49325o 

68 

01 

5io83o 

14 

47 

493040 

67 

999790 

.07 

67 

38 

506750 

i3 

48 

497078 
5oio8o 

66 

-S 

999786 

-07 

497293 

66 

76 

502707 

13 

49 

66 

999782 

.07 

501298 

66 

i5 

49S702 

II 

5o 

5o5o45 

65 

-48 

999778 

-07 

503267 

65 

55 

494733 

10 

5i 

8-508974 
512867 

64 

■89 

9-999774 

-07 

8-5o<>2oo 

64 

06 

11 -490800 

I 

52 

64 

•  31 

999769 

-07 

5i3ov>: 

64 

^9 

486902 

53 

516726 

63 

•7'^ 

999763 

•07 

5i6q6i 

63 

82 

483o39 

1 

54 

52o55i 

63 

•  19 

999761 

•07 

520790 

b3 

26 

479210 

6 

S5 

524343 

62 

-64 

999757 

.07 

5245S6 

62 

72 

473414  1  5 

56 

528102 

62 

-II 

999753 

-07 

523340 

62 

18 

47i65i 

4 

12 

531828 

61 

•  58 

999748 

-07 

532080 

6i 

65 

467920 

3 

535523 

61 

■  06 

999744 

•  07 

■535779 

61 

i3 

464221 

2 

59 

539186 

60 

•55 

999740 

•07 

539447 
543084 

60 

62 

40o553 

I 

60 

542819 

60 -04 

999735 

•  o-j 

60-12 

436916 

c 

Coeine 

D. 

Sine 

Cotang. 

D 

Tsuig 

(88    DEGREES.) 


20 


(2    DEGREES.)       A  TABLE   OF    LOGARITHMIC 


M. 

0 

Sico 

D 

Cosine 

D. 

Tang. 

D. 

Cotasg. 

8.542819 

60 

04 

9-999735 

07 

8-543084 

6o-I2 

11-456916  60 

I 

546422 

59 

55 

999731 

07 

546691 

^ 

62 

453309 

§ 

2 

540093 

59 

06 

999726 

07 
08 
08 

550268 

59 

14 

449732 

3 
4 

553539 
557054 

58 
58 

58 
11 

999722 
999717 

553817 
557336 

58 
58 

66 

53 

446183 
442664 

u 

5 

56o54o 

57 

65 

99971 3 

08 

560828 

57 

43917J   55 

6 

563999 

57 

19 

999708 

08 

564291 

57 

27 

433709  54 

I 

567431 

56 

74 

999704 

08 

567727 

56 

82 

432273 

53 

570836 

56 

3o 

999699 

08 

571137 

56 

38 

428863 

52 

9 

574214 

55 

87 

999694 

08 

574520 

55 

95 

425480 

5i 

10 

577366 

55 

44 

999689 

08 

577877 

55 

32 

422123 

5o 

II 

8-580892 

55 

02 

9-999685 

08 

8-581208 

55 

10 

1 1 -418792 
415486 

4o 

12 

584193 

54 

60 

999680 

08 

584514 

54 

68 

48 

i3 

587469 

54 

19 

999675 

08 

587795 

54 

27 

4l2205 

47 

U 

590721 

53 

79 

999670 

08 

59io5i 

53 

87 

408949 

46 

i5 

593948 

53 

39 

999665 

08 

594283 

53 

^0^ 

405717 

45 

i6 

597152 

53 

00 

999660 

08 

597492 

53 

4o25o8 

44 

n 

6oo332 

52 

61 

999655 

08 

600677 

52 

70 

399323 

43 

i8 

603489 

52 

23 

999650 

08 

6o3839 

52 

32 

396161 

42 

19 

606623 

5i 

86 

999645 

09 

606978 

5i 

94 

3o3022 

389906 

41 

20 

609734 

5i 

49 

999640 

09 

610094 

5i 

58 

40 

21 

8-612823 

5i 

12 

9-999635 

09 

8-613189 

5i 

21 

1I-3868II 

39 

22 

6i5Sqi 

5o 

76 

999629 

09 

616262 

5o 

85 

383738 

38 

23 

618957 

5o 

41 

999624 

09 

6i93i3 

5o 

5o 

380687 

ll 

24 

621962 

5o 

06 

999619 

09 

622343 

5o 

i5 

37765T 

25 

624965 

49 

72 

999614 

09 

625352 

49 

81 

374648 

35 

26 

627948 

49 

38 

099608 

09 

628340 

49 

47 

371660 

34 

27 

63091 1 

49 

04 

999603 

09 

63i3o8 

S 

10 

3686g2 

33 

28 

633854 

48 

7' 

999397 

09 

634256 

80 

365744 

32 

29 

636776 

48 

39 

999592 

09 

637184 

48 

48 

362816 

3i 

3o 

639680 

48 

06 

999586 

09 

640093 

48 

16 

359907 

3o 

3x 

8-642  563 

47 

75 

9-999581 

09 

8-642982 

47 

84 

11-357018 

20 

32 

645428 

47 

43 

999575 

09 

645853 

47 

53 

354147 

28 

33 

648274 

47 

12 

999570 

09 

648704 

47 

22 

351296 

27 

34 

65 1 102 

46 

82 

909504 

09 

65i537 

46 

9" 

348463 

26 

35 

65391 1 

46 

52 

99o558 

10 

654352 

46 

61 

345648 

25 

36 

656702 

46 

22 

999553 

10 

657140 

46 

3i 

342851 

24 

11 

659475 

45 

92 

999547 

10 

659928 

46 

02 

340072 

23 

662230 

45 

63 

999541 

10 

662689 

45 

73 

337311 

22 

39 

664968 

45 

35 

999535 

10 

665433 

45 

44 

334367 

21 

40 

667689 

45 

06 

999329 

10 

668160 

45 

26 

331S40 

20 

4: 

8-670393 

44 

79 

9-999524 

10 

8-670870 

44 

88 

11 -329i3o 

1^ 

42 

6730S0 

44 

5i 

999518 

10 

673563 

44 

61 

326437 

43 

675751 

44 

24 

999512 

10 

676239 

44 

34 

323761 

17 

44 

67S405 

43 

97 

999506 

10 

678900 

44 

17 

321100 

16 

45 

681043 

43 

70 

999500 

10 

681344 

43 

80 

3 18456 

i5 

46 

683665 

43 

44 

999493 
999487 

10 

684172 

43 

54 

3i5828 

14 

47 

686272 

43 

18 

10 

686784 

43 

28 

3i32i6 

i3 

48 

688863 

42 

92 

999481 

10 

689381 

43 

o3 

310619 

12 

49 

691438 

42 

67 

999475 

10 

69 1 963 

42 

77 

3o8o37  '  ?i 

5o 

693998 

42 

42 

999469 

10 

694329 

42 

52 

3o547i   10 

5i 

8  696543 

43 

'7 

g- 999463 

II 

8-697081 

42 

28 

11 -302919 

9 

52 

699073 

41 

92 

999456 

II 

699617 

42 

o3 

3oo383 

a 

53 

70 ! 589 

41 

68 

999450 

1 1 

702139 

41 

I^ 

297861 

1 

54 

704090 

41 

44 

999443 

11 

704646 

41 

53 

295354 

6 

55 

706577 

41 

21 

999437 

11 

707140 

41 

32 

292860 

5 

56 

709049 

40 

97 

999431 

11 

709618 

41 

08 

2Q0382 

4 

u 

7ii5o7 

40 

74 

999424 

u 

712083 

40 

85 

287917  '  3 

713952 
716383 

40 

5i 

999418  1 

11 

714534 

40 

62 

285465   2 

59 

40 

20 

999411  , 

11 

716972 

40 

40 

283028 

I 

60 

718800 

40-06 

999404 

II 

719396 

40-17 

280604 

M. 

r" 

Cosine 

D. 

Sine   1 

Cotatig. 

D. 

Ting. 

(87    PEGRKKS.) 


SINES   AND  TANGENTS.       (3   DEGREES.) 


21 


M. 

0 

Sine 

D. 

Cosine 

D. 

Tan?. 

I).   1 

(Jotanfj. 

8-718800 

40 -06 

9-999404 

-11 

8-719396 

40-17 

1 1  -  280604 

60 

I 

721204 

39-84 

999398 

-11 

721806 

39-95 

278194 

% 

a 

7235g5 

39-62 

999391 

-II 

724204 

39-74 

275796 

58 

3 

725972 

39-41 

999384 

-II 

726588 

39-52 

273412 

^I 

4 

728337 

39-19 

999378 

-  II 

728959 

39-30 

271041 

56 

5 

730688 

38-98 

999371 

•II 

73i3i7 

39-09 

268683 

5'j 

6 

733027 

38-77 

999364 

-12 

733663 

38-89 

266337 

54 

I 

735354 

38-57 

999357 

•12 

735996 

38-6S 

264004 

53 

737667 

38-36 

999350 

•  12 

738317 

38-48 

261683 

52 

9 

739969 

38-i6 

999343 

-12 

740626 

38-27 

239374 

5i 
5o 

10 

742209 

37-96 

999336 

-12 

742922 

38-07 

257078 

II 

3-744536 

37-76 

9-999329 

-12 

8-745207 

37-87 

11-254793 

^2 

12 

746802 

37-56 

999322 

•  12 

747479 

37-68 

252521 

48 

i3 

749055 

37-37 

9993 1 5 

•  12 

749740 

37-49 

25o26o 

47 

14 

751297 

37-17 

999308 

•  12 

751989 

37-29 

24801 1 

46 

i5 

753528 

36-98 

999301 

•  12 

754227 

37-10 

245773 

45 

i6 

755747 

36-79 

9992(^4 
999286 

-12 

756453 

36-92 

243547   44  1 

17 

757955 

36-61 

-12 

758668 

36-73 

24i332 

4.» 

i8 

76oi5i 

36-42 

999279 

•  12 

760872 

36-55 

239128 

42 

•9 

762337 

36-24 

999272 

-12 

763o65 

36-36 

236935 

41 

20 

76451 1 

36 -06 

999265 

-12 

760246 

36-i8 

234754 

40 

21 

8-766675 

35-88 

9 -999257 

-12 

8-767417 
76957^ 

36-00 

n-232583 

39 

22 

768828 

35-70 

999250 

•i3 

35-83 

23o422 

38 

23 

770970 

35-53 

999242 

-i3 

771727 

35-65 

228273 

37 

24 

773101 

35-35 

999235 

•i3 

773866 

35-48 

226134 

36 

25 

775223 

35-i8 

999227 

■  i3 

775995 

35-3i 

224oo5 

35 

26 

777333 

35-01 

999220 

-i3 

7781 i4 

35-14 

22I8S6 

34 

27 

779i34 

34-84 

999212 

-i3 

780222 

34-97 

219778 

33 

28 

781524 

34-67 

99g2o5 

•i3 

782320 

34-80 

217680 

32 

29 

7836o5 

34-51 

999197 

•  13 

784408 

34-64 

215592 

3! 

3o 

785675 

34-31 

999 1B9 

-i3 

786486 

34-47 

2i35i4 

3o 

3i 

8-787736 

34-18 

9-999181 

-i3 

8-788554 

34-3i 

II -21 1446 

=2 

32 

789787 
791828 

34-02 

999 '74 

-i3 

790613 

34- 15 

209387   20 

33 

33-86 

999 1 66 

.i3 

792662 

33-99 

207338  27 

34 

793859 

33-70 

999 '53 

-i3 

794701 

33-83 

205299  26 

35 

795S81 

33-54 

999 1 5o 

-i3 

796731 

33-68 

203269   25 

36 

797S94 

33.39 
33-23 

999142 

•  i3 

798752 

33-52 

201248  24 

ll 

799*^97 
801892 

999 '34 

.i3 

800763 

33-37 

199237   23 

33-08 

999 1 26 

-i3 

802765 

33-22 

197235   22 

39 

803876 

32-93 

999118 

•  13 

804758 

33-07 

195242   21 

40 

8o5852 

32-78 

9991 10 

•  i3 

806742 

32-92 

193258 

20 

41 

8-807819 

32-63 

9-999102 

•  i3 

8-808717 

32-78 

II-I91283 

10 

42 

809777 

32-49 

999094 

•14 

8io683 

32-62 

189317 

18 

43 

811726 

32-34 

9990S6 

-U 

81264I 

32-48 

187359 

n 

44 

813667 

32-19 

999077 

•14 

8145S9 

32-33 

185411 

16 

45 

8i55o9 

32-o5 

999069 

•14 

816529 

32-10 

1 8347 1 

i5 

46 

817522 

31-91 

99906 1 

•14 

818461 

32-o5 

181539 

14 

47 

819436 

3i-77 

999053 

•14 

820384 

3i-9i 

1 796 1 6 

i3 

48 

'   821343 

3i-63 

999044 

•14 

822298 

3i-77 

177702 

12 

49 

823240 

3i-49 
3i-3d 

999036 

•14 

824205 

3i-63 

175795 

II 

5o 

825i3o 

999027 

•14 

826103 

3i-5o 

173897 

10 

5i 

8-827011 

3l-22 

9-999019 

•14 

8-827992 

1  3i-36 

11-172008 

9 

8 

5j 

828884 

3i-oS 

999010 

•14 

829S74 

3i-23 

170126 

53 

'   830749 

30-95 
30-82 

999002 

•14 

831748 

3i- 10 

168252 

7 

-•4 

1   8J2601 

998993 

i  -14 

8336i3 

30-96 

166387 

6 

55 

834456 

30-69 

998984 

•14 

835471 

3o-§3 

164529 

5 

56 

836297 

3o-56 

998976 

•14 

837321 

30-70 

162679 

4 

% 

838i3o 

3o-43 

998967 

•  15 

839163 

3o-57 

160S37 

3 

830956 

3o-3o 

998958 

.i5 

840998 
842825 

3o-45 

159002 

2 

59 

841774 

30-17 

998950 

•15 

3o-32 

157175 

I 

to 

8435^5 

3o-oo 

998941 

-i5 

844644 

30-19 

155356 

0 

Coeine 

D. 

Sine 

Cotang. 

D. 

Tftng. 

M. 

(86    DEOREES.^ 


22 


(4    DEGREES.)      A  TABLE   OF   LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

60 

0 

8-843585 

3o-o5 

9-908941 

•i5 

8-844644 

3o- 19 

11 -155356 

I 

845387 

29-02 
29-80 

998932 

■i5 

846455 

30-07 

153545 

So 

1  is 

2 

84TI83 

99B923 

.i5- 

848260 

29-95 
29-82 

i5i74o 

J 

848971 

29-67 

998914 

•i5 

850007 

149943 

140154 

57 

4 

85075 1 

29-55 

99S905 

;  -15 

85 1 846 

29-70 

56 

5 

852525 

29-43 

998896 

•  15 

853628 

29-58 

146372  i  55 

6 

8:.429i 

29-3i 

998887 

•  15 

855403 

29-46 

144597  ;  54 

7 

RJO049 

2Q-19 

99'^878 

•i5 

857171 

29-35 

142820  1  53 

8 

b:)78oi 

29-01 

998^69 

-i5 

85S932 

29-23 

141068 

52 

9 

639546 

28-06 

998860 

-i5 

860686 

29- 1 1 

i3g3i4 

5i 

10 

861283 

28-84 

998851 

-15 

862433 

29-00 

137067 

1  5c 

II 

8-863oi4 

28-73 

9-998841 

.i5 

8-864173 

28-88 

11-135827 

49 

13 

864738 

28-61 

998S32 

-i5 

865906 

28.77 

134094 

48 

i3 

866455 

28-5o 

998823 

-16 

867632 

28-66 

132368 

47 

14 

8681 65 

28-39 

998813 

-16 

869301 

28-54 

1 30649 

46 

i5 

869868 

28-28 

998804 

.16 

871064 

28.43 

128936 

45 

i6 

871565 

28-17 

998795 

.16 

872770 

I   28-32 

127230 

44 

17 

873255 

28-06 

998785 

-16 

874469 

!  28.21 

125531 

43 

18 

874938 

27-95 

998776 

-16 

876162 

28-11 

123838 

42 

19 

876615 

27-86 

998766 

-16 

877849 

28-00 

I22r5i 

41 

20 

878285 

27-73 

998757 

.16 

879529 

27-89 

120471 

40 

21 

8-879949 

27-63 

9-998747 

•16 

8-881202 

27-79 
27-68 

11-118798 

^? 

32 

88 1 607 

27-52 

998738 

.16 

882869 

117131 

23 

883258 

27-42 

998728 

•16 

884530 

27-58 

115470 

3? 

24 

884903 

27-31 

998718 

•16 

886185 

27-47 

ii38i5 

36 

25 

886542 

27-21 

99S708 

.16 

887833 

27-37 

112167 

35 

26 

888174 

27-11 

998699 
9986S9 

•16 

889476 

27-27 

iio524 

34 

'7 

889S01 

27-00 

•16 

891 1 12 

27-17 

108888 

33 

28 

891421 

26-90 

998679 

•16 

892742 

27-07 

107258 

32 

29 

893035 

26-80 

998669 

•17 

894366 

26-97 

105634 

3i 

3o 

894643 

26-70 

998609 

•17 

895984 

26-87 

104016 

3o 

3i 

8-896246 

26-60 

9-998649 

•17 

8-897596 

26.77 

11-102404 

29 
28 

32 

807842 

26-51 

998639 

•17 

899203 

26-67 

100797 

33 

899432 

26-41 

998629 

•17 

900S03 

26-58 

099197 

27 

34 

901017 

26-31 

998619 

•17 

90230B 

26-48 

097602 

26 

35 

902596 

26-22 

998609 

•17 

9o39§7 

26-38 

096013 

25 

36 

904169 

26-F2 

998599 

•17 

905070 

26-29 

094430 

24 

3-7 

905736 

26-03 

99S589 

•17 

907147 

26-20 

092853 

23 

38 

907297 

25-93 

998578 

•17 

908719 
910280 

26-10 

001281 

089715 
o8Si54 

22 

39 

90S853 

25-84 

998568 

•17 

26-01 

21 

40 

910404 

25-75 

998558 

•17 

911846 

20-92 

20 

41 

8-911949 

25-66 

9-998548 

•17 

8-913401 

25-83 

11-086599 

19 

42 

913488 

25-56 

998537 

•17 

914951 

25-74 

085049 

IB 

43 

9l5022 

25-47 

998527 

•17 

916495 

25-65 

o835oO 

17 

U 

9i655o 

25-38 

998016 

•18 

qi8o34 

25-56 

081966 

16 

45 

918073 

25-29 

99S006 

-18 

919568 

20-47 

0S0432 

i5 

46 

919591 

25-20 

99S495 

•18 

921096 

25-38 

078904 

14 

s 

921103 

25-12 

998485 

-18 

922619 

25-30 

077381 

i3 

922610 

25-o3 

998474 

•18 

924136 

20-21 

075864 

12 

49 

924112 

24-94 

998464 

-18 

925649 

25-12 

074351 

11 

60 

925609 

24-86 

998453 

•18 

927156 

25-o3 

072844 

ID 

5i 

8-927100 

24-77 

9-998442 

-18 

8-928658 

24-9^ 

24 -S6 

11 -071342 

g 

52 

9285.S7 

24fc9 

998431 

•  18 

93oi5o 

069845 

53 

930068 

24-60 

99S421 

•i3 

931647 

24-78 

068353 

I 

54 

93 1 544 

24-52 

998410 

•  18 

933 1 34 

24-70 

066S66 

, 

55 

9330 1 5 

24-43 

998399 

•  18 

934616 

24-61 

065384 

5 

k 

56 

934481 

24-35 

998388 

•  18 

936093 

24-53 

063907 

4 

I 

u 

935942 

24-27 

993377 

■  18 

937555 

24-45 

062435 

3 

■ 

9373Q8 

24-19 

998366 

.18 

939032 

24-37 

060968 

2 

■ 

59 

938850 

24-11 

998355 

-18 

940404 

24-30 

0O9O06 

I 

■ 

60 

940296 

24-o3  1 

998344 

•18 

941902 

24-21 

o58o48 

0 

1 

I 

Cosine 

D.  ! 

Sine 

Cotang. 

D. 

Tang.  1  M.  j 

(85  DEGREES.) 


SIXES   AND   TANGENTS.       (5   DEGREES.) 


2? 


^ 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

'Til' 

24 -03 

9-998344 

.19 

8-94ig52 

24-21 

1 1  -  038048 

60 

I 

23-04 

99S333 

-19 

943404 

24-i3 

o56f)96 

59 
58 

3 

943174 

23-87 

99S322 

-19 

944832 

24-o5 

o55i48 

3 

944606 

23-79 

9983 1 1 

•  19 

946295 

23  •Q7 

033705 

^2 

4 

946034 

23.71 

998300 

•  19 

947734 

23 -gr 

o52  266 

56 

5 

947436 

23-63 

99S289 

•19 

949168 

23-82 

o5o832  j 

55 

6 

948874 
950287 

23-55 

998277 

•19 

930397 

23 -n 

049403  1 

54 

7 

23-48 

99S266 

•  19 

952021 

23-66 

047979 

53 

8 

95 1 696 

2J-40 

99S255 

•  19 

953441 

23 -60 

046559 

52 

9 

10 

93";  100 

23-32 

998243 

.19 

954836 

23-5i 

045144 

5i 

954499 

23-25 

998232 

-19 

956267 

23-44 

043733 

5o 

II 

8  955804 
937284 

23-17 

9-99'i22o 

-19 

8-957674 

23-37 

11-042326 

49 

12 

23-10 

998209 

-19 

939075 

23-2g 
23-23 

040(^25 

48 

i3 

938670 

23-02 

998107 
908186 

•  19 

960473 

o3g'j27 

S 

U 

960032 

22-95 

22-88 

■19 

90 1866 

23-14 

o38i34 

i5 

961429 

998174 

.19 

963255 

23-07 

o36745 

45 

i6 

962801 

22-80 

998163 

.19 

964639 

23-00 

o3536i 

44 

\l 

964 1 70 

22-73 

998151 

.19 

966019 

22-o3 

22-86 

o33g8i 

43 

965534 

22-66 

998130 

•  20 

967394 

o320o6 

42 

'9 

966893 
968249 

22-59 

998128 

•  20 

968766 

22-79 

o3i234 

41 

20 

22-52 

9981 16 

•  20 

970133 

22-71 

029867 

40 

21 

8-069600 

22-44 

9-99S104 

•  20 

S-971496 

22-65 

n -028504 

37 

22 

91'^') 'll 

22-38 

998002 

998080 
998068 

•  20 

972835 

22-57 

027145 

23 

972289 

22-3l 

•  20 

974209 

22-5l 

025791 

24 

973628 

22-24 

•  20 

975560 

22-44 

024440 

36 

25 

974962 

22-17 

99.S056 

•  20 

976906 

22-37 

023094 

35 

26 

076293 

22-10 

998044 

•  20 

978248 

22-3« 

02n32 

34 

s 

977619 

22 -03 

998032 

•  20 

979586 

22-23 

020414 

33 

32 

97.S941 

21-97 

998020 

•20 

980921 

22-17 

019079 

29 

980239 

21-30 

998008 

•20 

982231 

22-10 

017740 

016423 

3i 

3o 

981573 

21-83 

997996 

•iO 

983577 

22-04 

3o 

3i 

8-982883 

21-77 

9-997985 

•20 

3-984899 

21-97 

ii-oi5ioi 

29 

32 

984189 

21-70 

997972 

•20 

986217 

21-91 

013783 

28 

33 

9S5491 

21-63 

997959 

•20 

tli^^ 

21-84 

012468 

'Z 

34 

9S6789 
988083 

21-57 

997947 

•20 

988842 

21-78 

01 1 i58 

26 

25 

35 

21 -50 

997935 

•21 

990149 

21-71 

0008 3 1 

36 

989374 

21-44 

997922 

•21 

991451 

21-65 

008549 

24 

23 

ll 

990660 

21-38 

997910 

•21 

992750 

21-58 

007230 

991943 

2I-3l 

997'''97  1 

•21 

994045 

21-52 

005955 

22 

39 

993222 

21-25 

997885 

•21 

995337 

21-46 

004663 

21 

40 

994497 

21-19 

997872 

•21 

996624 

21-40 

003376 

20 

41 

8-995768 

21-12 

9-997860 

•2! 

8-997908 

21-34 

I J -002092 

19 

42 

9970J6 

21-06 

997847 

•21 

999188 

21-27 

000812 

18 

43 

998299 

21-00 

997835 

•21 

9-000465 

21-21 

10-999335 

7 

44 

999360 

20 -04 

997822 

•21 

001738 

21-l5 

998262 

16 

45 

9-000816 

20-87 

997809 

•21 

oo3oo7 

21-09 

996993 

i5 

46 

002060 
oo33i8 

20-S2 

997797 

-21 

004272 

21 -o3 

993728 

14 

47 

20-76 

997784 

-21 

003534 

20-97 

994466 

i3 

48 

004363 

20-70 

99777" 

-21 

006792 

20-QI 

993208 

12 

49 

op58o5 

20-64 

997738 

•21 

008047 
009298 

20 -83 

991953 

II 

5o 

007041 

20-58 

997745 

'21 

20-80 

990702 

ID 

5i 

9-008278 

20-52 

9.997732 

-21 

9-010546 

20-74 

ic -989454 

I 

52 

009310 

20-46 

997719 

-21 

01 1790 
oi3oJi 

20-68 

988210 

53 

010737 

20-40 

997706 

-21 

20-62 

986969 

7 

54 

01 1962 

20-34 

997693 

-22 

014268 

20-56 

985732 

6 

55 

0l3lH2 

20-20 

997680 

-22 

oi55o2 

2o-5i 

984498 

5 

56 

014400 

20-23 

997667 

•22 

0167)2 

20-45 

983268 

4 

0 

ll 

oi56i3 

20-17 

997654 

•22 

0179)9 

20-40 

982041 

3 

016824 

20-12 

997641 

•22 

019183 

20-33 

980817 

2 

59 

oi8o3i 

20-06 

997628 

■22 

02o4o3 

20-28 

970597 

I 

66 

019235 

20-00 

997614 

•22 

021620 

20-23 

978380 

0 

Cosine 

D. 

1   Sine 

'  Cotang. 

D. 

.  Tang-... 

M. 

(84  l;EOREKS.) 


24 


(.()    DEGREES.)       A    TABLE   OF     LOGARITHMIC 


M.' 

Sine      I 

X 

Cosine 

D. 

Tang. 

D. 

CotaLg.  { 

0 

9-019135   20 

•00 

9-997614 

•22 

9^02l620 

20-23 

10-978380 

6« 

I 

020435   19 

-95 

997601 

•  22 

022834 

20 

••7 

977166 

DO 

2 

02i632   19 

-89 

997588 

•22 

024044 

20 

•11 

973956 

58 

3 

022825   19 

-84 

997574 

•  22 

02525l 

20 

•  06 

9747.(9 

57 

4 

024016   19 

-78 

997561 

•  22 

026455 

20 

•  00 

973545 

56 

5 

025203   ig 

•73 

997547  j 

•  22 

027655 
028852 

19 

-95 

972345 

55 

6 

026386   19 

■  67 

997534 

•23 

19 

-90 

971148 

54 

I 

027567   19 

•62 

997520 

•23 

o3oo46 

•9 

-85 

960954 

53 

028744   19 

•57 

997507 

•23 

o3i237 

032423 

'9 

-79 

968763 

53 

9 

029918   19 

•5i 

9974o3 

•23 

>9 

-74 

967575 

5i 

10 

031089   19 

•47 

997480 

•23 

o336o9 

19 

-69 

966391 

5o 

II 

9032257   19 

•41 

9-997466 

•23 

9^034791 

19 

-64 

10-965209 

8 

12 

o3342i   19 

-36 

997452 

•23 

035969 

19 

-58 

96403 1 

i3 

034582   19 

3o 

997439 

■23 

037144 

'9 

-53 

962856 

47 

14 

035741   19 

25 

997423 

•23 

o383i6 

>9 

48 

961684 

46 

i5 

036896   1 9 

20 

99741 1 

-23 

039485 

19 

43 

9605 1 5 

45 

i6 

o38o48   19 

i5 

997397 

•23 

04065 1 

19 

38 

959349 

44 

'7 

039197   19 

10 

997383 

•23 

041813 

19 

33 

958187 

43 

i8 

040342    19 

o5 

907369 

•23 

042973 

19 

28 

957027 

42 

'9 

041485   18 

99 

997333 

•23 

044 I 3o 

19 

23 

955870 

41 

20 

042625   18 

94 

997341 

•23 

045284 

19 

18 

954716 

40 

21 

9-043762    18 

89 

9-997327 

•24 

9-046434 

19 

i3 

10-953566 

39 

38 

22 

044895   18 

84 

99731 3 

•24 

047382 

'9 

08 

952418 

23 

046026   18 

79 

697299 

•  24 

048727 

19 

o3 

951273 

37 

24 

047154   18 

75 

997283 

•24 

049860 

18 

98 

95oi3i 

36 

25 

048279   18 

70 

997271 

•24 

03 1008 

18 

93 

948992 
947836 

35 

26 

049400   18 

65 

997237 

•24 

o52i44 

18 

89 

34 

27 
28 

o5o5i9   18 

60 

997242 

•24 

053277 

18 

84 

946723 

33 

o5i63d   18 

55 

997228 

•24 

054407 

18 

79 

945593 

32 

29 

052749   18 

5o 

997214 

•24 

055535 

18 

74 

944465 

3i 

3o 

053859   18 

45 

997199 

•  24 

o56659 

18 

70 

943341 

3o 

3i 

9-054966   1 8 

41 

9.997185 

•24 

9-057781 

18 

65 

10-942219 

29 

32 

006071   18 

36 

997170 

•24 

058900 

18 

69 

941100 

28 

33 

057172   18 

3i 

997156 

•24 

060016 

18 

55 

939984 

27 

34 

058271   18 

27 

997141 

•24 

o6ii3o 

18 

5i 

93S870 

26 

35 

039367   18 

22 

997 '27 

•24 

062240 

18 

46 

937760 

25 

36 

060460   18 

17 

997112 

•24 

063348 

18 

42 

936652 

24 

37 

o6i55i   18 

i3 

99709S 

•24 

064453 

18 

37 

935547 

23 

38 

062639   18 

08 

997083 

•25 

065556 

18 

33 

934444 

22 

39 

063724   18 

04 

997068 

•25 

066655 

18 

28 

933345 

21 

40 

064806   17 

99 

997053 

•25 

067752 

18 

24 

93224S 

20 

41 

9-065885   17 

94 

9-997039 

•25 

9-068846 

18 

'9 

lo-93ii54 

\t 

42 

066962   17 

90 

86 

997024 

•25 

069933 

18 

IS 

930062 

43 

o68o36   17 

997009 

•25 

071027 

18 

ic 

928973 

>7 

44 

069107   17- 

81 

996994 

•25 

072113 

i8^ 

06 

927887 

16 

45 

070176   t-]- 

77 

996979 

•25 

073197 

18- 

03 

926803 

i5 

46 

071242   17- 

72 

996964 

•25 

074278 

17- 

97 

925722 

14 

47 

072306   17- 

68 

996949 

•23 

075356 

17- 

?3 

924644 

i3 

48 

073366   17- 

63 

996934 

•25 

076432 

17- 

§9 

923568 

13 

49 

074424   17- 

u 

996919 

•25 

o775o5 

«7- 

84 

922495 

11 

5o 

075480   17- 

996904 

•25 

078576 

n- 

80 

9214:4 

10 

5i 

9-076533   17- 

5o 

9-996889 

•25 

9-079644 

17- 

76 

10-920356 

n 

5j 

077583   17 
078631   17- 

46 

996874 

•25 

080710 

«7- 

72 

919290 

8 

53 

42 

996858 

-25 

081773 
082833 

17- 

67 

918227 

I 

54 

079676   17- 

38 

996843 

•25 

17  63 

917167 

55 

080719   17- 

33 

99^)828 

•25 

083891 

17  59 

916109 

5 

56 

081759   17- 

29 

996S12 

•  26 

084947 

17  55 

9i5o53 

4 

u 

082797   17- 

23 

996797 

■  26 

086000 

I7-5I 

9 1 4000 

3 

o838j2   17- 

21 

996782 

•  26 

087050 

17-47 
17-43 

912950 

3 

59 

084S64   1 7  - 

«7 

996766 

•  26 

088098 

911902 

I 

60 

085894   17- 

i3 

996751 

•26 

089144 

I7^38 

910856 

0 

Coaine     1) 

Si'ie 

1  Cotan^. 

D. 

Tang. 

M. 

(83    DEGREES.) 


SINES  AND  TANGENTS   (7  DEGREES.) 

2 

M. 

Sine 

D. 

Cosine 

D. 

Tang. 

1   D. 

Cotang. 

o 

9-085894 

17-13 

9-996751 

-26 

9-089144 

17-38 

io-9io856 

60 

I 

086922 

17.09 

996735 

-26 

090187 

17-34 

900813 

U 

3 

087947 

17-04 

996720 

-26 

091228 

17.30 

90S772 
907734 

3 

088970 

17-00 

996704 

-26 

092266 

17.27 

u 

4 

089990 

16  96 

996688 

-26 

093302 

17-22 

906698 

091008 

16.92 

996673 

-26 

094336 

17-19 
i7-i5 

905664 

55 

6 

092024 

16.88 

996657 

-26 

095367 

904633 

54 

I 

093037 

16.84 

99664 1 

.26 

096395 

17-11 

9o36o5 

53 

094047 

16-80 

996625 

.26 

097422 

17-07 

902578 

52 

9 

O95o56 

16.76 

996610 

-26 

098446 

17-03 

901554 

5i 

10 

096062 

16.73 

996594 

-26 

099468 

16-99 

900532 

5o 

II 

9-097065 

16.68 

9-996578 

•27 

9-100487 

16-95 

iD-8995i3 

49 

13 

098066 

16.65 

996562 

•27 

ioi5o4 

16-91 

898456 

48 

i3 

099065 

16.61 

996546 

•27 

I025i9 

i6-§7 

897481 

47 

14 

100062 

16.57 

996530 

•27 

103532 

16.84 

896468 

46 

i5 

ioio56 

16-53 

996514 

.27 

104542 

16.80 

895458 

45 

i6 

102048 

16-^9 

996498 

•27 

io555o 

16.76 

894450 

44 

\l 

io3o37 

16-45 

996482 

•27 

106556 

16-72 

893444 

43 

104025 

16-41 

996465 

•27 

107559 
io856o 

16-69 

892441 

42 

>9 

loSoio 

16-38 

996449 

•27 

16-65 

891440 

41 

20 

105992 

16-34 

996433 

•27 

109559 

16-61 

890441  1  40 

21 

9-106973 

i6-3o 

9-996417 

•27 

9-iio556 

16-58 

10-889444  '  39 
888449  3d 

22 

1 0795 1 

16-27 

996400 

•  27 

iii55i 

16-54 

23 

108927 

16-23 

•996384 

•  27 

112543 

i6-5o 

887457 

ll 

24 

1 0990 1 

16-19 

996868 

•27 

II3533 

16-46 

886467 

25 

110873 

16-16 

996351 

•27 

11 452 1 

16-43 

885479 
884493 

35 

26 

1 1 1842 

16-12 

996335 

•27 

ii55o7 

16-39 

34 

27 

1 1 2809 

16-08 

996318 

:U 

116491 

i6-36 

883509 

33 

28 

1 13774 

i6-o5 

996302 

1 17472 

16-32 

882528 

32 

29 

1 14737 

16-01 

996285 

-28 

118452 

16-29 
16-25 

881548 

3t 

3o 

115698 

15-97 

996269 

.28 

119429 

880571 

So 

3i 

9-II6656 

15.94 

9.996252 

-28 

9-120404 

16-22 

10.879596 

87S623 

ll 

32 

m6i3 

15:^7 

996235 

•  28 

121377 

16.18 

33 

II 8567 

996219 

•  28 

122348 

i6-i5 

877^52 

27 

34 

119519 

i5.83 

996202 

-28 

123317 

16-11 

876683 

i& 

35 

120469 

15.80 

996185 

-28 

124284 

16-07 

875716 

25 

36 

121417 

15-76 

996168 

.28 

125249 

16-04 

874751 

24 

ll 

122362 

i5-73 

9961 5i 

.28 

126211 

16-0! 

813789 

23 

i233o6 

15-69 

996134 

-28 

127172 

i5-97 

.S-,'2828 

22 

39 

124248 

i5-66 

9961 17 

-28 

i28i3o 

1 5 -94 

871870 

21 

40 

125187 

i5-62 

996100 

.28 

1 29087 

15-91 

870913  1  20 

41 

9-126125 

15-59 

9-996083 

•29 

9 -130041 

15-87 

10-869959 

19 

42 

127060 

i5-56 

996066 

-29 

I 30994 

i5-84 

869006 

18 

43 

127993 

i5-52 

996049 

-29 

i3iQ44 

i5-8i 

868o56 

\l 

44 

128925 

15-49 

996032 

-29 

132893 
133839 

15-77 

867 1 07 

45 

129854 

15-4D 

996015 

•29 

15-74 

866161 

i5 

46 

130781 

i5-42 

995998 
995980 

•29 

134784 

15-71 

865216   14 

S 

131706 

15-39 
15-35 

•29 

135726 

15-67 

864274 

i3 

i3263o 

993963 

•  29 

136667 

i5-64 

863333 

13 

49 

i3355i 

i5-32 

995946 

•29 

137605 

i5-6i 

862395 

II 

5o 

134470 

15-29 

995928 

•29 

138542 

i5-58 

861438   10 

5i 

9-135387 

i5.25 

9-99591 1 

-29 

9.139476 

15-55 

io.86o524   9 

52 

i363o3 

l5-22 

995894 

•29 

140409 

i5.5i 

859591 

8 

53 

137216 

15-19 

995876 

•29 

141340 

15.48 

858660 

I 

54 

i38i28 

i5.i6 

995859 

•  29 

142269 

15.45 

857731 
856804 

55 

139037 

l5.I2 

995841 

-29 

143196 

i5-42 

5 

56 

139044 

15-09 

995823 

•29 

144121 

i5-39 
15-35 

855879 

4 

ll 

i4o85o 

i5-ot) 

995806 

•29 

145044 

854956 

3 

141754 

i5-o3 

995788 

••9 

145966 

15.32 

854034 

2 

59 

142655 

i5oo 

995771 

-29 

146885 

15.29 

853ii5   1 

60 

143555 

14-96 

995753 

■29 

147803 

i5-26 

852197   0 

C!oeine 

D. 

Sine 

Cot.ani?.  1 

D. 

Tmg.       M. 

(82  1 

>KOF 

BBS.) 

Is 


26 


(8  DEGREES  )   A  TABLE  OF  LOGARITHMIC 


M. 

0 

Sine 

D. 

Cosine 

D. 

Taug. 

D. 

Cotflr.g. 

9.1J3555 

14-96 

9  995753 

•3o 

9-147803 

15-26 

10  852197 

60 

I 

144453 

14-93 

995735 

•3o 

148718 

1 5 

-23 

8512,82 

S 

a 

I4534Q 

14-90 

995717 

•3o 

149632 

i5 

-20 

85o368 

3 

146243 

14-87 

995699 

•3o 

I 5o544 

i5 

•'7 

849456 

57 

4 

I47I36 

14-84 

995681 

•  30 

i5i454 

i5 

-14 

848546 

56 

5 

148026 

14-81 

995664 

•3o 

152363 

i5 

-11 

847637 

55 

6 

148015 
149802 

14-78 

995646 

•3o 

153269 

i5 

-08 

846731 

54 

I 

14-75 

993628 

•3o 

154174 

i5 

-o5 

845826 

53 

i5o686 

14-72 

9956 1 0 

•3o 

i55o7- 

i5 

-02 

844923 

52 

9 

i5i569 

14-69 

995591 

-3o 

155978 

14 

•99 

844022 

5i 

10 

i5245i 

14-66 

995573 

•3o 

156877 

14 

-96 

843123 

5o 

11 

9-1 53330 

14-63 

9-995555 

•  3o 

9-157775 

14 

93 

10-842225 

49 

12 

1 54208 

i4-6o 

995537 

-3o 

1 5867 1 

14 

-90 

841329 

48 

i3 

i55o83 

14-57 

995519 

•  3o 

1 59565 

14 

•87 

840433 

47 

14 

155957 

14-54 

995501 

•  3, 

160457 

14 

-84 

839543 

46 

i5 

1 56830 

14-51 

9954S2 

•3i 

161347 

14 

.81 

838653 

45 

i6 

157700 

14-48 

995464 

•3i 

162236 

1-4 

-79 

837764 

44 

\l 

1 58569 

14-45 

995446 

-3i 

i63i23 

14 

76 

836877 

43 

159435 

14-42 

995427 

•3i 

164008 

14 

•73 

835992 

42 

19 

i6o3oi 

14-39 

99.5409 

•3i 

164892 

14 

70 

835 108 

41 

20 

161164 

14-36 

995390 

-3i 

165774 

14 

67 

834226 

40 

21 

9-162025 

14-33 

9-995372 

.31 

9-166654 

14 

64 

10-833346 

39 
38 

22 

162885 

i4-3o 

995353 

.31 

167532 

14 

61 

832468 

23 

163743 

14-27 

995334 

-3i 

168409 

14 

58 

83i59i 

u 

24 

164600 

14-24 

995316 

•  3i 

169284 

14 

55 

830716 

25 

165454 

14-22 

995297 

-3i 

170157 

14 

53 

829843 

35 

26 

i663o7 

14-19 

995278 

-3i 

171029 

14 

5o 

828971 

34 

27 

23 

i6J^oo8 

i4-i6 

995260 

•  3i 

171899 

14 

47 

828101 

33 

14-13 

995241 

.32 

172767 

14 

44 

827233 

32 

29 

168856 

14-10 

995222 

•32 

173634 

14 

42 

826366 

3i 

3o 

169702 

14-07 

995203 

-32 

174499 

14 

39 

825501 

3o 

3i 

9-170547 

i4-o5 

9-995184 

-32 

9-175362 

14 

36 

10-834638 

29 

32 

•  ,171389 

14-02 

995 1 65 

•32 

176224 

14 

33 

823776 

28 

33 

'172230 

13-99 

995146 

.32 

177084 

14 

3i 

822916 

27 

34 

173070 

13-96 

993127 

.32 

177942 

14 

28 

822058 

26 

35 

173908 

13-94 

995108 

•32 

178799 

14 

25 

821201 

25 

36 

174744 
175578 

13-91 

995089 

•32 

1796DD 

14 

23 

820345 

24 

ll 

13-88 

995070 

•32 

i8o5o8 

14 

20 

819492 

23 

17641 1 

13-86 

•  995o5i 

•32 

i8i36o 

14 

\l 

818640 

22 

39 

177242 

T3-83 

095o32 

•32 

182211 

14 

817789 

21 

40 

178072 

^3-8o 

99501 3 

•32 

i83o59 

14 

12 

8 '6941 

20 

41 

9-178900 

13-77 

9-9949Q3 

•32 

9-183907 

14 

09 

10-816093 

ll 

42 

\llll' 

13-74 

994974 

•32 

184752 

14 

07 

815248 

43 

13-72 

994955 

•32 

185597 
186439 

14 

04 

8i44o3 

n 

44 

181374 

13-69 

994935 

•32 

14 

02 

81 3561 

16 

45 

182196 

i3-66 

9949 1 6 

33 

187280 

i3 

99 

812720 

i3 

46 

j83oi6 

13-64 

994896 

•  33 

188120 

i3 

96 

8118S0 

14 

S 

183834 

i3-6i 

994877 

-33 

188958 

i3 

93 

81 1042 

i3 

18465 1 

i3-59 

994857 

•  33 

189794 

i3 

810206 

12 

49 

185466 

i3-56 

994838 

•  33 

190629 

i3 

89 

809371 

11 

5o 

186280 

i3-53 

994818 

■33 

191462 

i3 

86 

8o8538 

10 

5i 

9-187092 

i3-5i 

9-994798 

•  33 

9-192294 

i3 

84 

10-807706 

I 

52 

187903 

13-48 

994779 

•  33 

193124 

i3 

81 

806876 

53 

1887 1 2 

13-46 

994759 

•  33 

193953 

i3 

79 

806047 

7 

54 

189519 
190325 

i3-43 

994739 

•  33 

194780 

i3- 

76 

8o522o 

6 

55 

i3-4i 

994719 

.33 

193606 

i3- 

74 

804394 

5 

56 

191130 

13-38 

994700 

.33 

196430 

i3- 

n 

803570 

4 

u 

191933 

13-36 

994680 

•33 

197253 

i3- 

8o;747 

3 

193734 

13-33 

994660 

•  33 

198074 

i3- 

66 

801926 

2 

59 

193534 

i3-3o 

994640 

•  33 

198894 

i3 

64 

801 106 

I 

60 

194332 

13-28 

994620 

•  33 

199713 

i3-6i 

800287 

0 

Cosine 

D.   1 

Sine 

1  Cot;ing. 

D. 

Tancr. 

M. 

(81  DKGBEE8.) 


SINES  AND  TANGENTS.   (9  D 

EGREE.) 

27 

u. 

Sine   I   l 

>. 

Coeine 

D. 

Tang.  ; 

D. 

Cotang. 

0 

9-194332   i3 

28 

9-994620 

•33 

9-'997i3 

i3-6i 

10-800287 

6c 

I 

196129   i3 

26 

994600 

•  33 

200629 

13-69 

799471 
798655 

U 

a 

195920   i3 

23 

994580 

.33 

201 343 

13-56 

3 

196719   i3 

21 

994560 

34 

202169 

13-64 

797841 

^1 

4 

197511   i3 

18 

994540 

•34 

202971 

i3-62 

797020 
796218 

56 

5 

198302   i3 

16 

994519 

•34 

203782 

i3-49 

55 

6 

10909 1   i3 

i3 

994499 

•34 

204592 

i3-47 

796408 

54 

7 

199879   1 3 

11 

994479 

•34 

206400 

i3-45 

"94600 

53 

8 

200666   1 3 

08 

994459 
994438 

•34 

206207 

i3-42 

793793 

52 

9 

20i45i   i3 

06 

•34 

207013 

i3-4o 

792987 
792183 

61 

10 

202234    i3 

04 

994418 

■34 

207S17 

13-33 

5o 

II 

9-2o3oi7   i3 

01 

9-994397 

•34 

9-208619 

i3-35 

10-791381 

40 

12 

203797   12 

99 

994377 

•34 

209420 

13-33 

700680 
780780 

48 

i3 

204577   12 

96 

Q94357 

•34 

210220 

i3-3i 

47 

14 

205354   12 

94 

994336 

•34 

211018 

13-28 

788982 

46 

i5 

2o6i3i   12 

tl 

9943 1 6 

•34 

211816 

13-26 

788186 

45 

i6 

206906   1 2 

994295 

•34 

212611 

i3-24 

787389 
786695 

44 

17 
I^ 

207679   12 

87 

904274 

•35 

21 3406 

l3-21 

43 

208452   12 

85 

994254 

•35 

214108 
214989 

i3-i9 

786802 

42 

'9 

209222   12 

82 

994233 

•35 

13-17 

786011 

41 

20 

209992   12 

80 

9942 1 2 

•  35 

216780 

i3-i6 

784220 

40 

21 

9-210760   12 

78 

9-994191 

•35 

9-216668 

l3-12 

10-783432 

It 

22 

21 1 526   12 

75 

99417' 

•  35 

2n35o 

i3-io 

782644 

23 

2:2:9!   12 

73 

994 I 5o 

•35 

218142 

i3-o8 

781868 

37 

24 

2i3o55   12 

71 

994129 

•  35 

218926 

i3-o6 

781074 

36 

25 

2i38(8   12 

68 

994 1 08 

•  35 

219-710 

i3-o3 

780290 

35 

26 

214579   12 

66 

994087 

•  35 

220492 

i3-oi 

770608 
778723 

34 

3 

215333   12 

64 

994066 

•  35 

221272 

12-99 

33 

216007   12 

61 

994045 

•35 

222032 

12-97 

777948 

32 

29 

216854   12 

59 

994024 

.35 

222830 

12-94 

777170 

3i 

3o 

217609   12 

57 

994oo3 

•35 

2  236o6 

12-92 

776394 

3o 

3i 

9-218363   12 

55 

9-993981 

•35 

9-224382 

12-90 

io^7756i8 

52 

32 

219I16    12 

53 

993960 

•35 

223156 

12-88 

774844 

33 

219S68   12 

5o 

993939 

.35 

226929 

12-86 

774071 

27 

34 

220618   12 

48 

993(^1 8 
993 S96 

•35 

226700 

12-84 

773300 

26 

35 

221 367   12 

46 

•36 

227471 

12-81 

772629 

26 

36 

223Il5    12 

44 

993875 

•  36 

228239 

12-79 

771761 

24 

ll 

222861     12 

42 

993854 

•  36 

229007 

12-77 

770993 

23 

2236o6    12 

39 

993832 

•  36 

229773 

12-76 

770227 

22 

39 

224349    12 

37 

993811 

•  36 

23o639 

12-73 

769461 
768698 

21 

4o 

225092    12 

35 

993789 

•  36 

23i3o2 

12-71 

20 

4i 

9-225833   12 

33 

9-993768 

•36 

9-232066 

12-69 

10-767935 

\l 

42 

226573    12 

3i 

99^746 

•36 

232826 

12-67 

767174 

43 

227311     12 
228048     12 

28 

993725 

•36 

233586 

12-65 

766414 

17 

44 

26 

993703 

•  36 

234346 

12-62 

765656 

16 

45 

228784    12 

24 

993681 

•  36 

235 io3 

12-60 

764897 

i5 

46 

229518    12 

-22 

993660 

•36 

235869 

12-68 

764141  '  i4 

47 

23o252    12 

-20 

993638 

•  36 

2366 14 

12-56 

763386   i3 

48 

230984    1 2 

•  18 

993616 

•36 

237368 

12-54 

762632   12 

49 

231714    12 

-16 

993594 

•  37 

238120 

12-62 

761880   II 

5o 

232444    12 

•14 

993572 

•37 

238872 

12 -60 

761 ;28 

10 

5i 

9-233172    12 

-12 

9-993550 

.37 

9-239622 

12-48 

i: -760378 

I 

53 

233899  :   12 

•09 

993528 

•37 

24o3ti 

I2^46 

769629 

75.8882 

53 

53462:  1   12 

-07 

993506 

•37 

241118 

12-44 

7 

54 

235349     12 

•  o5 

993484 

•37 

241866 

12-42 

768136 

6 

55 

236073     12 

-o3 

993462 

•  37 

242610 

12-40 

767390 

5 

56 

236795    12 

•  01 

993440 

•  37 

243354 

12-38 

766646 

i  4 

57 

2375i5   11 

•99 

993418 

•37 

244097 

12-36 

766903 

3 

58 

238235   11 

•97 

993396 

•  37 

244839 

12-34 

766161 

2 

59 

238953   11 

-95 

993374 

•37 

245679 

12-33 

754421 

I 

6o 

239670   1 1 

•93 

993351 

•37 

246319 

ia-3o 

753681 

0 

1 
1  . 

Ck)6ine  |   ] 

0. 

Sinfi 

Cotang. 

D. 

Tang. 

M. 

(80    DEGREES.) 


28 


(10    DEGREES.)      A   TABLE   OF    LOGARITUMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

60 

0 

9-239670 

n-93 

9-993351 

•37 

9^2463i9 

i2-3o 

10-753681 

I 

240386 

II. 01 

993329 

•37 

247057 

12-28 

752943 

?1 

a 

241 lOI 

11.69 

993307 

•37 

247794 

12-26 

7' 2206 

3 

241814 

11.87 

993285 

•37 

248530 

12-24 

751470  57 

4 

242526 

11-85 

9^3262 

-37 

249264 

12-22 

750736   56 

5 

243237 

n-83 

993240 

■M 

249998 
2507J0 

12-20 

750002   55 

6 

243947 

11-81 

993217 

12-18 

7492 --o  ;  54 

m 

244656 

11-79 

993195 

-38 

25i46i 

12-17 

748539 

53 

» 

245363 

11-77 

993.172 

-38 

252191 

I2-I5 

747809 

52 

9 

246069 
246775 

11-75 

993 '49 

•  38 

252920 

12-13 

747080 

5i 

10 

11.73 

993127 

•38 

253648 

12-11 

746352 

5o 

II 

9-247478 

11-71 

9-993104 

•  38 

9-254374 

12-09 

10-745626 

S 

12 

248181 

11-69 

993081 

•  38 

255 1 00 

12-07 
12-03 

744900 

i3 

248883 

nil 

993059 

-38 

255824 

744176 

47 

14 

J49583 

993o36 

-38 

256547 

12-03 

743453  i  46 

li 

250282 

11-63 

99301 3 

•  38 

257269 

12-01 

742731  1  45 

i6 

250980 

11-61 

992990 

•  38 

257990 
258710 

12-00 

742010 

44 

'7 

251677 

\\:U 

992967 

•38 

11.98 

741290 

43 

i8 

252373 

992944 

•  38 

259429 

11.96 

740571 

42 

«9 

253067 

11-56 

992021 

.38 

260 1 46 

11-94 

739854 

41 

20 

253761 

11-54 

992898 

•38 

260863 

11-92 

739137 

4c 

21 

9-254453 

11-52 

9-992875 

•  38 

9-261578 

II  -oo 

10-738422 

?i 

22 

255144 

11.50 

992852 

•  38 

262292 

11-89 

737708 

23 

255834 

11.48 

992829 

•39 

263oo5 

11-87 

11-85 

736995 

37 

24 

256523 

11-46 

992806 

•  39 

263717 

736283 

36 

i5 

25721 1 

11-44 

992783 

•  39 

26.1428 

11-83 

735572 

35 

26 

257898 
258583 

11-42 

992759 

•39 

265 1 38 

ii-8i 

734862 

34 

11 

11-41 

992736 

•  39 

265847 

11-70 

734153 

33 

259268 

11-39 

992713 

•39 

266555 

11-78 

733445 

32 

29 

259951 

11-37 

992690 

.39 

267261 

11-76 

732739 
732033 

3i 

3o 

260633 

11-35 

992666 

•39 

267967 

11-74 

3o 

3i 

9-26i3i4 

11-33 

9-992643 

.39 

9-268671 

11-72 

10-731329 

29 

J2 

261994 

ii-3i 

992619 

.39 

269375 

11-70 

730623 

28 

33 

262673 

ii-3o 

992596 

•39 

270077 

11-69 

729923 

27 

34 

263351 

11-28 

992572 

•39 

270779 

;::S 

729221 

26 

35 

264027 

11-26 

992549 

•  39 

271410 

728521 

25 

36 

264703 

11-24 

992523 

•39 

272178 

11-64 

727822 

24 

37 

263377 

11-22 

992501 

.39 

272876 

11-62 

727124 

23 

38 

26605 I 

11-20 

992478 

•40 

273573 

11-60 

726427 

22 

39 

266723 

11-19 

992454 

40 

274269 

11-58 

725731 

21 

40 

267395 

II-I7 

992430 

•  40 

274964 

11-57 

725o36 

20 

41 

9.268065 

II-I5 

9 •992406 

•40 

9.275658 

11.55 

10-724342 

19 

42 

268734 

11-13 

992382 

•40 

276351 

11-53 

723649 

18 

43 

269402 

ii-ii 

992359 
992335 

•  40 

277043 

ii.5i 

722957 

>7 

44 

270069 

II-IO 

•40 

277734 

ii-5o 

722266 

16 

45 

270730 

1 1 -08 

99231 1 

-40 

278424 

11-48 

7215-6 

i5 

46 

271400 

11-06 

992287 

-40 

279113 

11-47 

720887 

14 

47 

272064 

ii-o5 

992263 

•40 

27980! 
280488 

11-45 

720199 

i3 

48 

272-'26 

II -03 

992239 

•40 

11-43 

71q5i2 

13 

49 

273388 

II-OI 

992214 

•40 

281,74 

11-41 

718826 

U 

5o 

174049 

10-99 

992190 

•40 

28i858 

11-40 

718142 

10 

5i 

9-274708 

10-98 

9-992166 

.40 

9-282542 

11-38 

10-717458 

I 

52 

275367 

10-96 

992142 

•  40 

283225 

11-36 

716775 

53 

276024 

10-94 

992117 

•41 

283907 
284588 

11-35 

716093 

1 

54 

276681 

10-92 

99  1093 

•41 

11-33 

715412 

6 

55 

277337 

10-91 

992069 

.41 

285268 

11-31 

714732 

5 

56 

277991 

10-89 

992044 

•41 

285947 

ii-3o 

714053 

4 

u 

278644 

10-87 
10-86 

992020 

•41 

286624 

11-28 

713376 

3 

279297 
279048 
280399 

991996 

•41 

287301 

11-26 

7 1 2699 
712023 

2 

59 

10.84 

991971 

•41 

287977 

11-25 

I 

60 

10  83 

991947 

•41 

288652 

11-23 

711348 

0 

Coeinfi 

D. 

Sine   ! 

Cotang. 

D. 

Tang. 

M. 

(Y9  DEGREES.) 


BINES  AJsD   TAXGENl^.      (11    DEGREES.) 


29 


VL. 

Sine 

D. 

Cosine  | 



D. 

Tang, 

D. 

Cotatig. 

-■  T 

0 

9' 280699 

10-82 

9  0Qi9-«7 

•41 

9.288662 

II. 23 

10  711348 

5o 

I 

281248 

10 

81 

991922 

•41 

2S9326 

11-22 

710674 

59 
58 

a 

281897 

10 

79 

991897 

-41 

289999 

11-20 

7 1 000 1 

3 

382044 

10 

77 

991873 

-41 

29067 1 

11.18 

709329 

57 

4 

38J190 
383836 

10 

76 

9Q1848 

•41 

291342 

11-17 

708668 

50 

5 

10 

74 

991823 

•41 

392013 

ii-i5 

707987 

55 

6 

384480 

10 

72 

091709 

•41 

292682 

11-14 

707318 

54 

I 

385i24 

10 

71 

991774 

•42 

293350 

11-13 

706660 

53 

385766 

10 

69 

901749 

•42 

294017 

II-II 

706983 

52 

9 

386408 

10 

67 

991724 

•42 

294684 

11-09 

7033 16 

5i 

10 

387048 

10 

66 

991699 

•42 

293349 

11-07 

704661 

5o 

II 

0-287687 

10 

64 

9-991674 

•42 

9-296013 

11-06 

10-7039^7 

49 

48 

la 

288326 

10 

63 

991649 

■42 

296677 

11-04 

703323 

i3 

388964 

10 

61 

991624 

•42 

297339 

11 -03 

702661 

47 

14 

389600 

10 

5o 

991399 

•42 

298001 

II-OI 

701999 

46 

i5 

390236 

10 

58 

991574 

•42 

298662 

11-00 

70i338 

45 

i6 

390870 

10 

56 

991549 

•42 

299322 

10-98 

700678 

44 

\l 

291304 

10 

54 

991624 

•42 

299980 

.0-96 

700020 

43 

292137 

292768 

10 

53 

991498 

•42 

3oo638 

10-96 

699362 
698706 

42 

«9 

10 

5i 

991473 

■42 

301296 
301961 

10-93 

41 

30 

293399 

10 

5o 

991448 

■42 

10-93 

698049 

40 

31 

9 •294029 

10 

48 

9-991422 

•42 

9-302607 

10.90 

10-697393 

39 

33 

294658 

10 

46 

991397 

42 

3o326i 

10.89 

696739 

38 

33 

295286 

10 

45 

991372 

-43 

3o3oi4 
304667 

10.87 

6960S6 

37 

34 

393013 
396539 

10 

43 

991346 

43 

10-86 

696433 

36 

35 

10 

42 

991321 

43 

3o52i8 

10-84 

694782 

35 

36 

297164 

10 

40 

991295 

43 

306869 

10-83 

694131 

34 

^l 

397788 

10 

39 

991270 

43 

306619 

io-8i 

693481 

33 

38 

398412 

10 

37 

99 » 244 

43 

307168 

10-80 

692832 

32 

?9 

399034 

10 

36 

991 2 18 

43 

307816 

10-78 

6921S5 

3i 

3o 

399655 

10 

34 

991193 

43 

3o8463 

10-77 

691637 

3o 

3r 

9-300276 

10 

32 

9-991167 

43 

9-309109 

10-75 

10-690891 

S 

33 

300S95 

10 

3i 

991141 

43 

309764 

10-74 

690246 

33 

3oi5i4 

10 

29 

991115 

43 

310398 

10-73 

689602 

27 

34 

302l32 

10 

28 

991090 

43 

3iio42 

10-71 

688968 

26 

35 

302748 

10 

26 

991064 

43 

3ii686 

10-70 

68831 5 

35 

36 

3o3364 

10 

25 

99io38 

43 

812327 

10-68 

687673 

34 

ll 

3o3(27q 

10 

23 

991012 

43 

312967 

10-67 

687033 

23 

304393 

10 

22 

990986 

43 

3i36o8 

10-65 

686392 

22 

39 

303207 

10 

20 

990960 

43 

314247 

10-64 

686753 

31 

40 

3o58i9 

10 

19 

990934 

44 

314885 

10-63 

686116 

20 

41 

9 -306430 

10 

17 

9-990908 

44 

9  !i5523 

10-61 

10-684477 

1'^ 

43 

307041 

10 

16 

990S82 

44 

316169 

10.60 

683841 

43 

307630 

10 

14 

990855 

44 

316796 

10.53 

683206 

'7 

44 

308239 

10 

i3 

990829 

44 

317430 
318064 

10-57 

682670 

16 

45 

308867 

10 

II 

990803 

44 

10-55 

681936 

i5 

45 

309474 
3 10080 

10 

10 

990777 

44 

318697 

1064 

68i3o3 

14 

47 

10 

08 

990760 

44 

319329 

10. 53 

680671 

i3 

48 

3io685 

10 

07 

990724 

44 

319961 

10. 5i 

68oo3o 
670408 
676778 

:a 

49 

311289 

10 

o5 

990697 

44 

320392 

10. 5o 

II 

5o 

311893 

10 

04 

990671 

44 

321222 

10-48 

10 

5i 

9-312495 

10 

o3 

9-990644 

44 

9-32i85i 

10-47 

10-678149 

I 

53 

3 1 3097 

10 

01 

990618 

44 

322479 

10.45 

677521 

53 

313698 

10 

00 

990391 

44 

323io6 

10-44 

676S94 

I 

54 

314297 

9 

98 

990666 

44 

323733 

10-43 

676267 

55 

314897 

9 

97 

990638 

44 

324358 

10-41 

676642 

5 

56 

3 1 5495 

9 

96 

990611 

45 

3249S3 

10.40 

676017 

4 

ll 

316092 
316689 

9 

94 

9904S5 

45 

326607 

10-39 

674393 

3 

9 

93 

990468 

45 

326231 

10-37 

673769 

a 

59 

317284 

9 

91 

990431 

45 

326853 

10-36 

673147 

1 

6o 

317879 

9.90 

9904)4  1 

45 

327475 

10-35 

673625 

0 

Cosine 

D. 

Sine 

Cotanor. 

D. 

Tan?.  , 

J^. 

(78  degkeeb) 


30' 


(12    DEGREES.;      A  TABLE   OF   LOGARITniflC 


M. 

Sine 

D. 

Cosine 

D. 

Tan?. 

D. 

Cotnng.  \ 

o 

g. 317870 
318473 

9.00 

9-88 

9-990404 

•45 

9-327474 

10-35 

10-672526 

60 

I  . 

990378 

•45 

328095 

10 

33 

671905 

5o 

2 

019066 

9-87 

9go35i 

•45 

328713 

10 

32 

671285 

58 

3 

319658 

9-86 

990324 

•45 

329334 

10 

3o 

670666  57  I 

4 

320249 

9-84 

990297  1 

-45 

329953 

10 

29 

670047 

56 

5 

320840 

9-83 

990270 

•45 

33o370 

10 

28 

669430 
6688  rv 

55 

6 

32i43o 

9-82 

990243 

•45 

331187 

10 

26 

54 

7 

322019 

9-80 

990215 

•45 

33i8o3 

10 

25 

668197 

53 

8 

322607 

9-79 

990188 

•45 

332418 

10 

24 

bb-]^^2 

5s 

9 

323194 

9-77 

990 161 

•45 

333o33 

10 

23 

666067 

5i 

10 

323780 

9.76 

990134 

•43 

333646 

10 

21 

666354 

5o 

II 

9  324366 

9-75 

9-990107 

.46 

9-334259 

10 

20 

10-665741 

49 

12 

324950 

9-73 

990079 

.46 

334871 

10 

19 

665 129 

48 

i3 

325d34 

9-72 

999032 

■46 

335482 

10 

'7 

664518 

47 

14 

326117 

9-70 

990025 

989997 

.46 

336093 

10 

16 

663907 

46 

i5 

326700 

9.69 

•46 

336702 

10 

i5 

663298 
6626*^9 

45 

i6 

327281 

9-68 

989970 

•16 

337311 

10 

i3 

44 

\l 

327862 

9-66 

989942 

•i6 

337919 

10 

12 

662081 

43 

328442 

9-65 

989915 

989A87 

989S60 

•46 

338327 

10 

1 1 

661473 

42 

19 

329021 

9-64 

•46 

339133 

10 

10 

660867 

41 

20 

329399 

9-62 

•46 

339739 

10 

08 

660261 

40 

21 

9-330176 

9-61 

9-989832 

•46 

9-340344 

10 

07 

10-659656 

39 

22 

330753 

9-60 

989804 

•46 

340948 

341332 

10 

06 

659032 

38 

23 

33i329 

9-58 

9^9777 

•46 

10 

04 

658448 

37 

24 

33 1903 

9.57 

989749 

•47 

342155 

10 

o3 

657845 

36 

25 

332478 

9-56 

989721 

•47 

342757 

10 

02 

657243 

35 

26 

33303I 

9-54 

989693 

•47 

343358 

10 

00 

656642 

34 

11 

333624 

9-53 

98q665 

•47 

343938 

9 

^ 

656042 

33 

334195 

9-52 

989637 

•47 

344558 

9 

655442 

32 

29 

334766 

9 -50 

989609 

•47 

343157 

9 

97 

654843 

3i 

3o 

335337 

9.49 

989382 

•47 

343755 

9 

96 

654245 

3o 

3i 

9-335906 

9-48 

9.989553 

•47 

9-346353 

9 

94 

10-653647 

^2 

32 

336475 

9-46 

989525 

•47 

346949 

9 

93 

653o3i 

2S 

33 

337043 

9-45 

989497 

•47 

347545 

9 

92 

652455 

27 

34 

337610 

9-44 

989469 

•47 

34S141 

9 

9' 

65i859 

26 

35 

338176 

9-43 

989441 

•47 

348735 

9 

90 

88 

651265 

25 

36 

338742 

9-41 

989413 

•47 

349329 

9 

650671 

24 

U 

339306 

9-40 

989384 

•47 

349922 

9 

87 

650078 

23 

339871 

9-39 

989356 

•47 

33o5i4 

9 

86 

640486 

23 

39 

340434 

9-37 

089328 

•47 

331 106 

9 

85 

648894 

21 

40 

340996 

9-36 

989300 

•47 

351697 

9 

83 

6483o3 

20 

41 

9-341558 

9-35 

9-989271 

•47 

9-352287 

9 

82 

io-6477i3 

\l 

42 

342119 

9-34 

9R9243 

•47 

352876 

9 

81 

647124 

43 

342679 

9-32 

9'^92'4 

••47 

353465 

9 

80 

646535 

n 

44 

343239 

9-3i 

9S9186 

•47 

354053 

9 

79 

645947 
645360 

:6 

45 

343797 

9-3o 

989157 
989128 

•47 

334640 

9 

■'I 

i5 

46 

3443d5 

9-29 

.48 

355227 

9 

76 

644773 

14 

s 

344912 

9-27 

989 1 00 

.48 

35581 3 

9 

75 

644187 

.3 

345469 

9-26 

989071 

.48 

356308 
356982 

9 

^ 

643602 

13 

49 

346024 

9-25 

989042 

•  48 

9 

73 

643018 

II 

5o 

346579 

9-24 

989014 

•48 

357366 

9 

•71 

642434 

10 

5i 

9-347134 

9-22 

9-988985 

.48 

9-358i49 

9 

70 

10  64i85i 

I 

52 

3476S7 

9-21 

988936 

•  48 

358731 

9 

69 

641269 

53 

348240 

9-20 

988927 

.48 

359313 

9 

68 

640687 

7 

54 

34S792 

9.19 

988898 

•48 

359S93 

9 

67 

640107 

6 

55 

3493^*3 

9-17 

988869 

•48 

360474 

9 

66 

630 J  j6 

5 

56 

349^93 

9-16 

988840 

.48 

36io53 

9 

65 

638947 
638368 

4 

u 

35o443 

9-i5 

988811 

•49 

36i632 

9 

63 

3 

350992 

9-14 

988782 

•49 

362210 

9 

62 

637790 

3 

59 

35i34o 

9-i3 

988753 

•49 

362787 

9 

61 

637213 

1 

66 

352088 

9-11 

988724 

•49 

363364 

9-60 

636636 

a 

1  Cosino 

D. 

Sine 

Cotanq:. 

D. 

TKEg. 

(77   DEGREES.) 


SINES  AND  TANGENTS 

.   ^13  DEOREE9.) 

81 

tlL 

Sine 

D. 

Cosine 

D. 

Tang.  1 

D. 

Cotang, 

0 

9.3.12088 

9-1.  1 

9-9'*''i724 

•49 

9-363364 

9.60 

io-636o36 

60 

1 

352035 

9- 10  1 

988695 

49 

363940 

9-59 

630060 

s 

1 

353 181 

9-00 
9-o3 

988666 

■49 

364315 

9-58 

635485 

3 

353726 

988636 

-49 

365o90 

9-57 

634910 

57 

4 

354271 

9.07 

988607 

•49 

365664 

9-55 

.  634336 

56 

5 

354^(5 

9-o5 

988578 

•49 

366237 

9  ■  54 

633763 

55 

t 

3  5 '.353 

9-o4 

988548 

•49 

366810 

9-53 

633190 

54 

- 

355901 

9-o3 

98S519 

•49 

867382  1 

9-52 

632(i8 

53 

i 

356443 

9-02 

988489 

•49 

367Q53 

9-5i 

632C47 

52 

9 

356984 

9-01 

988460 

•40 

368524 

9-5o 

631476 

5i 

10 

357524 

8-99 

988430 

■49 

369094 

9-49 

630906 

5o 

II 

9.358064 

8-93 

9-988401 

•49 

9-369663 

9-48 

io-63o337 

49 

12 

3586o3 

8-97 

98H,}7i 

•49 

370282 

9-46 

629768 

48 

i3 

359141. 

8-96 

988342 

.49 

370799 

9-45 

629201 

47 

U 

359678 

8-95 

988312 

•  5o 

371367 

9.44 

62^633 

46 

i5 

36021 5 

8-93 

9882S2 

•  5o 

371933 

9-43 

628067 

45 

i6 

360752 

8-92 

988252 

•  50 

372499 

9-42 

627501 

44 

\l 

361287 

8-91 

988223 

•  50 

373064 

9-41 

626936 

43 

35 1 82 2 

8-90 
8-89 
8-83 

988193 

•  5o 

373629 
374193 

9-40 

626371 

42 

'9 

362156 

988163 

.5o 

9-3o 

625307 

41 

20 

362089 

98S133 

•  So 

374736 

9-3i 

625244 

40 

31 

9-363432 

8.87 

9-988103 

•5o 

9-375319 

9-37 

10-624681 

39 

22 

363954 

8-85 

Q^So73 

•5o 

375881 

9-35 

624119 
623558 

38 

23 

364485 

8-84 

983043 

-5o 

376442 

9-34 

37 

24 

35  )iii6 

8-83 

988013 

.5o 

377003 

9-33 

622997 

36 

23 

3  15546 

8-82 

987983 

-5o 

377563 

9-32 

6224)7 

35 

26 

366075 

8-81 

9879)3 

.5o 

378122 

9.31 

621.S78 

34 

11 

3656o4 

8-80 

•  9'^7>22 

•  So 

378681 

Q-3o 

621 3 19 

33 

367i3i 

8-79 

987892 

.5o 

379289 

9-29 

620(61 

32 

29 

36765c) 

8.77 

987862 

-So 

379"'97 

9-28 

62020] 

3i 

3o 

368i85 

8-76 

987832 

.51 

38o334 

9-27 

619646 

3o 

3i 

9-36871 1 

8.75 

9-987801 

.51 

9-380910 

926 

10-61 9090 

29 

32 

369236 

8-74 

9^777' 

.5i 

381466 

9-25 

6i8534 

28 

33 

369761 

8 -.73 

987740 

.5i  , 

382020 

9-24 

6i-''}8o 

27 

34 

370285 

8-72 

987710 

.5i 

3'<2575 

9-23 

6174^5 

^t 

35 

370808 

8-71 

987679 

•  5i 

383129 

9-22 

616871 

25 

36 

37i33o 

8-70 

9f^7('49 

.5i 

383682 

9-21 

6i63i8 

24 

ll 

371852 

8-69 

987618 

.51 

384234 

9-20 

6157O6 

23 

372373 

8-67 

987533 

.5i 

384786 

9.IC 

Si52i4 

22 

39 

372S9', 

8-66 

987557 

.5i 

385337 
385888 

9-i8 

61466I 

21 

40 

373414 

8-65 

987526 

.5i 

9.17 

614112 

20 

41 

9  373933 

8-64 

9-987496 

.5i 

9-386438 

9.. 5 

10  61 3562 

\l 

42 

374452 

8-63 

987465 

•  51 

38698-» 

9-14 

6i3oi3 

43 

374970 

8-62 

987434 

.5i 

387336 

9.13 

612464 

17 

44 

375487 

8-6i 

987403 

-52 

388084 

9.12 

61 1916 

16 

45 

376003 

8-6o 

987372 

.52 

388631 

9.11 

611369 

i5 

46 

3765 IQ 

8-59 
8-58 

987341 

.52 

339178 

9.10 

610822 

14 

8 

377035 

987310 

.52 

339724 

9-00 

610276 

i3 

377549 

8-57 

987279 

.32 

390270 

9-08 

609730 

12 

49 

378063 

8-56 

987248 

.52 

390815 

9.07 

609185 

1 1 

5o 

378577 

8-54 

987217 

.52 

391360 

9-06 

608640 

10 

5i 

9-379089 

8-53 

9-9871S6 

.52 

9-391903 

9-o5 

10-608097 
607553 

I 

52 

379601 

3-52 

987 1 55 

.52 

392447 

9-04 

53 

38oii3 

8-5i 

987124 

•52 

392989 

9-o3 

607011 

7 

54 

330624 

8-5o 

987092 

.52 

393331 

9-02 

606469 

6 

55 

33 11 34 

8-49 
8-48 

987061 

.52 

394073 

9-01 

6o5q27 

5 

56 

381643 

987030 

■52 

394614 

9-00 

6o5386 

4 

ij 

3821 52 

8-47 

986998 

.52 

395154 

ill 

604846 

3 

382661 

8-46 

986967 

.52 

395604 
396233 

6o43o6 

2 

59 

383i68 

8-45 

986936 

•52 

8-97 

603767    I 

60 

383675 

8-44 

986904 
1  Sine 

•52 

1 

1 

396771 

8-96 

1 

603229  1  0 

Cosine 

I). 

i  Cotan^. 

1   D. 

Taiii?.  I  M. 

27 


(70    DEGREES.) 


i2 

ri4 

DEGIIEES.)   A  ' 

TABLE  OF  LC 

)GARITH 

lMIC 

M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Coiang. 

0 

9.383675 

8.44 

9.986904 

•52 

9-596771 

8-96 

10-603229 

6.3 

I 

384182 

8-43 

986873 

.53 

397309 

8-96 

602691  j 
602164  1 

t3 

a 

384687 

8-42 

986841 

•  53 

397846 

8-95 

3 

383192 

8-41 

986809 

.53 

398383 

8-94 

601617 

57 

4 

385697 

8-40 

986778 

•  53 

398919 

8-g3 

6orjSi 

56 

5 

386201 

8-3g 

986746 

.53 

399455 

8-g2 

600545 

55 

6 

386704 

8-38 

986714 

•  53 

399990 

8-gi 

600010 

54 

I 

387207 
387709 

8-37 
8-36 

986fe83 
986651 

•  53 
•53 

4oo524 
4oio58 

8-90 

599476 
5g8g42 

53 

52 

9 

388210 

8-35 

9S6619 

.53 

401591 

8-83 

5984og 

5i 

10 

3887 II 

3-34 

986587 

.53 

402124 

8-87 

597876 

5o 

II 

9-389211 

8-33 

n. 986555 

•  53 

9.402656 

8-86 

10-597344 

^2 

48 

12 

3897 1 1 

8-32 

986523 

.53 

403187 

8-85 

596S13 

i3 

390210 

8.3i 

986401 

.53 

403718 

8-84 

5g62S2 

47 

14 

390708 

8-3o 

986439 

•  53 

404249 

8-83 

5g575i 

46 

i5 

391206 

8-28 

986427 

•  53 

404778 

8-82 

5g5222 

45 

i6 

391703 

8.27 

986395 

•53 

4o53o8 

8-81 

5g46gj 

44 

17 

392199 
3g26g5 

8-26 

986363 

•54 

4o5836 

8-80 

594164 

43 

i8 

8-25 

986331 

•54 

4o6364 

8-79 

593636 

42 

'9 

393191 
393685 

8-24 

986299 

•54 

4o68g2 

8-7^ 

5g3io8 

41 

20 

8-23 

986266 

•54 

407419 

8-77 

5g258i 

40 

21 

9-394179 

8-22 

9-986234 

•54 

9-407945 

8-76 

10-592055 

39 

22 

394673 

8-21 

986202 

•54 

408471 

8-75 

5gi529 
5gioo3 

38 

23 

395166 

8-20 

9S6169 

•54 

408997 

8-74 

11 

24 

395658 

8-19 
8-18 

986137 

•54 

409321 

8-74 

590479 
5^9955 

25 

3961 5o 

986104 

•54 

410045 

8-73 

35 

26 

396641 

8-17 

986072 

•54 

4i(i56g 

8.72 

58943. 

34 

11 

397132 

8-17 

986039 

•54 

411092 

8-71 

58S908 

33 

397621 

8-16 

986007 

•54 

4ii6i5 

8-70 

588385 

32 

29 

39S111 

8-i5 

985974 

•54 

412137 

8-6g 

587S63 

3i 

3o 

398600 

8-14 

985942 

•54 

412658 

8-63 

587342 

3o 

3i 

9-399088 

8-i3 

9-9S5009 

.55 

9-4 i3 179 

8-67 

10-586821 

29 

32 

399575 

8-12 

985876 

•  55 

4i36gg 

8-66 

586301 

28 

33 

400062 

8-11 

985843 

•  55 

4i42ig 

8-65 

5857S1 

27 

34 

400549 
4oio33 

8-10 

985811 

•  55 

414738 

8-64 

585262 

26 

35 

8-09 

985778 

.55 

4i5257 

8-64 

584143 

25 

36 

401 520 

8-08 

985745 

•  55 

415773 

8-63 

5842:25 

24 

37 

4o2oo5 

8-07 

985712 

•  55 

4i62g3 

8-62 

583707 

23 

38 

402489 

8-06 

980679 

•  55 

416810 

8-6i 

583190 

21 

39 

402972 

8-o5 

983646 

.55 

417326 

8-60 

582674 

21 

40 

403455 

8-04 

9856 1 3 

.55 

417842 

8-59 

582158 

20 

41 

9-403938 

8-o3 

9-985580 

.55 

9-418358 

8-58 

iD-58i642 

10 

18 

42 

404420 

8-02 

985547 

-55 

418873 

8-57 

581127 

43 

404901 

8-01 

985514 

.55 

41938- 

8-56 

58o6i3 

17 

44 

4o5382 

8-00 

985480 

.55 

419901 

8-55 

58oogg 
57g585 

16 

45 

4o5862 

7 -99 

985447 

.55 

42041 5 

8-55 

i5 

46 

406341 

7-98 

985414 

•  56 

420927 

8-54 

37Q073 

14 

47 

406820 

7-97 

985380 

-56 

42 1 440 

8-53 

578560  ,-3  1 

48 

407299 

7-96 

985347 

-56 

421912 

8-52 

578048  1  .2  ^ 

49 

407777 

7-95 

985314 

-56 

422463 

^^' 

577337 

11 

5^ 

40*^254 

7-94 

985280 

.56 

422974 

8-5o 

577026 

10  ' 

5i 

9.408731 

7-94 

9-985247 

.56 

9-423484 

8-49 

10-576516 

Q 

52 

409207 

7-93 

985213 

•  56 

1   4239g3 

8-48 

576007 

8 

53 

401^682 

7-92 

985180 

•  56 

1   4243o3 

8-48 

575497 

7 

54 

410157 

7-91 

985146 

•  56 

425oii 

8-47 

574989 

6 

55 

410632 

]X 

9851 13 

•  56 

425519 

8-46 

574481 

5 

56 

4 1 1 1 06 

980079 
985043 

•  56 

426027 

8-45 

573973  1  4  1 

U 

411579 

7-83 

.56 

426534 

8-44 

573466 

i 

4i2o52 

7-87 

98501 1 

.56 

427041 

8-43 

572930 
J172453 

2 

5q 

412524 

7-86 

984978 

•  56 

427547 

8-43 

I 

60 

412996 

7-85 

984944 

•  56 

428052 

8-42 

'J71948  !  0 

Coaine 

D. 

Siue 

1  Cotaiig. 

D. 

Tang. 

M. 

(75 

DKGF 

tsus.) 

SINES  AND  TANGENTS.     (15  DEGREES.) 


33 


M. 

Sine 

D. 

Cosine 

1  ^• 

Tang. 

D. 

Cotai\g. 

1 

( 

60 

1 

0 

9-412996 

7-85 

9-984944 

•p 

9-428053 

8.42 

10-571948 

I 

413467 

7-84 

984910 

•57 

428557 

8.41 

571443 

59 

3 

413938 

-83 

984876 

■V 

429062 

8^40 

5  ;o938 

58 

3 

414408 

•83 

984842 

•57 

429566 

8.39 
8.38 

570434 

57 

4 

414878 

-82 

984808 

•37 

430070 

569930 

56 

5 

415347 

-81 

984774 

•57 

430573 

8.38 

569427 

55 

6 

4i58i5 

-80 

984740 

•57 

431073 

8.37 

568925 

54 

i 

416283 

•79 

984706 

•57 

43 1577 

8-36 

568423 

53 

416751 

•78 

9S4672 

■^ 

432079 

8-35 

567921 

52 

9 

417217 

•77 

984637 

•57 

432580 

8^34 

567420 

5i 

10 

417684 

-76 

984603 

•57 

433o8o 

8-33 

566920 

5o 

II 

9-4i8i5o 

•75 

9-984569 

•57 

9-433580 

8.32 

10-566420 

S 

12 

4i86i5 

74 

984533 

•57 

434080 

8.32 

565920 

i3 

419079 

73 

984500 

•57 

434379 

8.3i 

565421 

47 

14 

419544 

73 

984466 

•57 

435078 

8.3o 

564922 

46 

i5 

420007 

72 

984432 

.58 

435576 

8.29 

564424 

43 

i6 

420470 

71 

984397 

•  58 

436073 

8-28 

563927 

44 

■I 

420933 
42i3o5 

70 

984363 

-58 

436370 

8.28 

563430 

43 

18 

^2 

984328 

-58 

437067 

8-27 

562933 

42 

Is) 

42i8di 
4223i8 

68 

984294 

•  58 

437563 

8-26 

562437 

41 

20 

67 

984259 

•58 

438059 

8-25 

561941 

40 

21 

9-422778 

67 

9-084224 

.58 

9-438554 

8-24 

10.561446 

s 

32 

423238 

6(i 

9S4190 

.58 

439048 

8-23 

560952 

33 

423697 

65 

98405 

.58 

439543 

8-23 

560457 

37 

34 

424136 

64 

984120 

•  58 

44oo36 

8-22 

559964 

36 

25 

424615 

63 

984085 

•  58 

440529 

8-21 

559471 

35 

36 

425073 

62 

984050 

•  58 

441022 

8-20 

55S978 
5584^6 

34 

11 

425530 

61 

984015 

.58 

44i5i4 

b-19 

33 

425987 

60 

983981 

.58 

442006 

8-19 

557994 

32 

»9 

426443 

60 

983946 

•  58 

442497 

8-i8 

5575o3 

3i 

3o 

426899 

59 

9S3911 

.58 

442988 

8-17 

557012 

3o 

3i 

9-427354 

58 

9-983875 

.58 

9-443479 
443968 

8-16 

10-556521 

20 

33 

427809 

57 

983840 

.59 

8.16 

556o32 

28 

33 

428263 

56 

983805 

.59 

444458 

8.i5 

555542 

27 

34 

428717 

55 

983770 

.59 

444947 

8-14 

555033 

26 

35 

429170 

54 

983735 

•  59 

445435 

8-i3 

554565 

25 

36 

429623 

53 

9S3700 

•  59 

445923 

8. 12 

554077 
553589 

24 

3? 

430075 

52 

983664 

•  59 

44641 1 

8-12 

23 

33 

43o527 

52 

983629 

•  59 

446898 

8-11 

553 102 

22 

39 

430978 

5i 

983594 

.59 

447384 

8-10 

552616 

21 

40 

431429 

DO 

983538 

.59 

447870 

8-09 

552 i3o 

20 

41 

9-431879 

49 

9-983523 

.59 

9-448356 

8-09 

10-551644 

]l 

42 

432329 

49 

983487 

.59 

448841 

8-08 

55ii59 

43 

432778 

48 

983452 

.59 

449326 

8-07 

550674 

17 

44 

433226 

47 

983416 

.59 

449810 

8-06 

550190 

16 

43 

433675 

46 

983381 

•  59 

430294 

8-06 

649706 

i5 

46 

434122 

45 

983345 

.59 

430777 

8-03 

54Q223 

14 

s 

434569 

44 

9833og 

.59 

431260 

8-04 

548740 

i3 

435oi6 

44 

983273 

•  60 

431743 

8-o3 

548257 

13 

p 

435462 

43 

983238 

•  60 

432225 

8-02 

547775 

li 

5o 

435908 

7-42 

983202 

•60 

452706 

8-02 

547294 

10 

5i 

9-436353 

7  41 

9-983166 

•  60 

9^453i87 

8-01 

10-546813 

t 

53 

436798 

7  40 

983 1 3o 

•60 

453668 

B-00 

546332 

53 

437242 

7-40 

983094 

.60 

454148 

7^99 

545852 

7 

54 

437686 

7-39 

983od8 

.60 

454628 

7^99 

545372 

6 

55 
56 

438129 

7-38 

983022 

.60 

453107 

7-98 

544893 

5 

438572 

7-37 

982986 

.60 

455586 

7^97 

544414 

4 

u 

439014 

7-36 

982950 

•  60 

456064 

7-96 

543936 

3 

439456 

7-36 

982914 

.60 

450542 

7-96 

543458 

1 

^ 

^l 

7-35 

982878 

•60 

457019 

7-95 

542981 
54a  304 

I 

6o 

7-34 

982842 

.60 

457496 

7-94 

0 

Cosine  1 

D. 

Sipo 

Cotang. 

D. 

__Taus^_ 

M.J 

iU 

DKCR 

SEhi.) 

__ — «— 1 

S4 


(16    DEGKEES.)      A  TABLE   OF   LOGARITHMIC 


M. 

0 

Sine 

U. 

Cosine 

D. 

TaiLg. 

D. 

CoUing. 

Ao 

9-440338 

7-34 

9-982842 

•60 

9-45749^ 

7-94 

10^ 542504 

I 

440778 

7-?3 

982805 

•  60 

.457973 

93 

542027 

h 

2 

441218 

7-32 

982769 

•  61 

458449 

93 

54i55i 

f 

3 

441658 

7-3i 

982733 

-61 

438923 

92 

5410-3 

57 

4 

442096 

7-3. 

982696 

•6i 

459400 

9' 

540600 

56 

5 

442535 

7-3o 

982660 

•61 

459aT5 

90 

540125 

53 

6 

44S';73 

7-29 

982624 

•61 

46o34q 

r, 

539631 

54  ! 

I 

443410 

7-28 

982587 

•61 

460823 

539177 

53 

44384T 

7-27 

982551 

•61 

461297 

88 

538703 

52 

9 

444-?84 

7-27 

982514 

-61 

461770 

88 

538230 

5i 

10 

444720 

7-26 

982477 

•61 

462242 

7  »7 

537758 

5o 

II 

9-445i55 

7-25 

9-982441 

•61 

0-462714 

7.86 

10-537286 

49 

12 

445590 

7-24 

982404 

•61 

463 186 

85 

536814 

48 

i3 

446025 

7-23 

982367 

-61 

463658 

85 

536342 

47 

14 

446459 

7-23 

982331 

•61 

464129 

84 

535871 

46 

i5 

446^93 

7-22 

982294 

-61 

464399 

83 

535401 

45 

i6 

447336 

7-21 

982237 

•61 

465069 

83 

534931 

44 

\l 

447759 

7-20 

9S2220 

-62 

465530 

82 

534461 

43 

448191 

7-20 

982183 

-62 

466008 

81 

533992 
533524 

42 

19 

448623 

7-i8 

982146 

-62 

i66476 

80 

41 

20 

449054 

982109 

•62 

466945 

80 

533o55 

40 

21 

9-449485 

7-16 

9-982072 

-62 

9-467413 

79 

io^532587 

39 

22 

449Qi5 

982035 

•62 

467880 

7» 

532120 

38 

23 

45o345 

7-16 

981998 

•62 

468347 

78 

53 1653 

37 

24 

430773 

7-15 

9S1961 

-62 

468814 

77 

53 1186 

36 

25 

45i2o4 

7-14 

981924 

-62 

469280 

76 

530720 

35 

26 

45i632 

7-.3 

981886 

-62 

469746 

73 

53 02 54 

34 

27 

452060 

7  13 

981849 

-62 

4702 1 1 

75 

529789 

33 

28 

452488 

7  12 

981812 

-62 

470676 

74 

529324 

32 

2g 

452oi5 

7-II 

981774 

-62 

471141 

73 

528859 

3i 

3o 

453342 

7-10 

98.737 

•62 

471605 

73 

52839? 

3o 

3i 

9-453768 

7-10 

9-981699 

-63 

9^472068 

72 

io^527932 

20 

23 

32 

454194 

7-OQ 

7 -08 

981662 

•63 

472532 

7« 

527468 

33 

454619 

981625 

-63 

472995 

71 

527005 

27 

34 

455o44 

7-07 

981587 

-63 

473437 

70 

526543 

26 

33 

455469 

7-07 

98 1 549 

•63 

473919 

69 

526081 

j5 

36 

455893 

7-06 

981512 

-63 

474381 

69 

523619 

24 

]l 

4563 16 

7-o5 

981474 

•63 

474842 

68 

525i58 

23 

456739 

7 -04 

981436 

-63 

4753o3 

67 

524697 

22 

39 

457162 

7 -04 

981399 

•63 

473763 

67 

524237 

31 

40 

457584 

7-o3 

981361 

-63 

476223 

66 

523777 

20 

41 

9 -458006 

7-02 

9-981323 

•  63 

9-476683 

65 

io^5233i7 

;s 

42 

458427 

7-01 

981285 

•  63 

477'42 

65 

522858 

43 

458848 

7-01 

981247 

•  63 

477'JO' 

64 

522399 

\i 

44 

459268 

7-00 

981209 

•63 

478059 

63 

521941 

45 

4596S8 

6-99 

9S1171 

•  63 

478517 

63 

521483 

i5 

46 

460 1 08 

6-98 

981133 

•64 

478975 

62 

521023 

14 

4- 

460527 

6-98 

9810(^5 

•  64 

479432 

61 

520368 

i3 

48 

460946 

6-97 

981037 

•  64 

479889 

61 

52011 1 

la 

49 

461 364 

6-96 

981019 

•64 

480343 

60 

519655 

II 

5o 

461782 

6-95 

980981 

•  64 

4S0801 

-59 

519199 

10 

5x 

9-462199 

6-95 

9-980942 

•  64 

9-481257 

-59 

ic-5i874^ 

I 

5a 

462616 

6-94 

980904 

•  64 

481712 

•  58 

518288 

53 

463o32 

6-93 

980866 

•64 

482167 

-57 

517833 

I 

54 

463448 

6-93 

980827 

•  64 

482621 

17 

517379 

55 

463864 

6-92 

980789 

•64 

483075 

-56 

516925 

5 

56 

464279 

6-91 

980750 

•  64 

483529 

■  55 

516471 

4 

s 

464694 

6-eO 

9807 1 2 

•  64 

483982 

■  55 

5i6oi8 

3 

465 108 

6-00 

980673 

•  64 

484435 

-54 

5 1 5563 

2 

59 

465522 

6-89 

980635 

•  64 

484887 

•53 

5151;^ 

« 

60 

465935 

6-88 

980596 

.64 

485339 

7-53 

5i4«-i 

0 

1  Cosine 

i   D. 

Sine 

Cotanff. 

D. 

Tai^. 

(73  1 

DEUR 

.EES.) 

SINES  AND   TANGENTS,       (17    DEGREES.) 


86 


M. 

0 

Sine 

D.   1 

1 

Cosine 

D. 

Tang.  1 

D. 

Cotiing. 

9-465935 
466348 

6-88 

9-980596 

-64 

9-485339 

7-55 

10-514661 

60 

I 

6-88 

98o5d8 

-64 

485791 

7-52 

514209 

u 

3 

466761 

6-87 

980019 

■65 

486242 

7.51 

5i3758 

.  3 

467173 

6-86 

9804S0 

-C5 

486693 

7-5i 

5i33o7 

57 

4 

467535 

6-85 

980442 

-65 

487143 

7-5o 

512857 

56 

* 

467996 
468407 

6-85 

980403 

•65 

487593 

7-49 

512407 

55 

6 

0-84 

980364 

•65 

488043 

7-49 

511957 

54 

7 

468817 

6-83 

980325 

•65 

488492 

7-48 

SinoS 

53 

8 

469227 

6-83 

980286 

•65 

488941 

7-47 

5iio59 

52 

9 

469637 

6-8f 

980247 

•65 

489390 
489833 

7-47 

5io6io 

5i 

10 

470046 

6-81 

980208 

•  65 

7-46 

510162 

5o 

II 

9-470455 

6-8r) 

9-980169 

-65 

9-490286 

■j-aS 

10-509714 

S 

12 

470S63 

6.8j 

980130 

-65 

490733 

7-45 

509.267 
508820 

i3 

471271 

6-70 

980091 

-65 

491180 

7-44 

47 

14 

471679 

6.7§ 

980052 

-65 

491627 

7-44 

508373 

46 

i5 

472086 

6-78 

980012 

•  65 

492073 

7-43 

507927 

45 

i6 

472492 

6-77 

979973 

•  65 

492319 

7.43 

507481 

44 

11 

473898 

6-76 

979934 

•  66 

492965 

7-42 

507035 

43 

473304 

6-76 

979895 

•  66 

493410 

7-41 

5o63QO 

42 

19 

473710 

6-75 

979835 

•  66 

493854 

7-40 

506146 

41 

20 

4741 i5 

6-74 

979816 

-66 

494299 

7-40 

5o57oi 

40 

21 

9-474519 

6-74 

9-979776 

•  66 

9-494743 

7-40 

io-5o5257 

39 

22 

474923 

6-73 

979737 

•  66 

495186 

7-39 

504814 

38 

23 

475327 

6.72 

979697 
97965S 

•  66 

49563o 

7.38 

504370 

37 

24 

473730 

6-72 

•  66 

496073 

7-37 

503927 

36 

23 

476133 

6-71 

979618 

•  66 

4g65i5 

7-37 

5o3485 

35 

26 

476536 

6-70 

979579 

•  66 

496957 

7-36 

5o3o43 

34 

s 

476938 

6-69 

979539 

•  66 

497399 

7-36 

5o26oi 

33 

477340 

6-69 

979499 
979459 

•  66 

497841 

7-35 

5o2i59 

32 

29 

477741 
478142 

6-6§ 

•  66 

498232 

7-34 

501718 

3i 

3o 

6-67 

979420 

•  66 

498722 

7^34 

501278 

3o 

3i 

9-478542 

6-67 

9-979380 

•  66 

9-499>63 

7.33 

io-5oo837 

29 

32 

478942 

6-66 

979340 

•  66 

499603 

7-33 

5oo397 
499958 

23 

33 

479342 

6-65 

979300 

•  67 

5ooo42 

7^32 

^I 

34 

479741 

6-65 

979260 

•  67 

500481 

7-31 

499519 

26 

35 

480140 

6-64 

979220 

•  67 

500920 

7-31 

499080 

25 

36 

480539 

6-63 

979180 

•67 

5oi359 

730 

498641 

24 

37 
3^ 

480Q37 

6-63 

979140 

•  67 

501707 

502233 

7-30 

498203 

23 

481334 

6-62 

979100 

.67 

7-20 
7.28 

497765 

22 

39 

48 1 73 1 

6.61 

979059 

-67 

502672 

497328 

21 

40 

482128 

6-61 

979019 

-67 

5o3 1 09 

7-a8 

496891 

20 

4i 

9-482525 

6-60 

9-978979 

•  67 

9 -503546 

7-27 

10-496454 

'2 

42 

482021 

6-59 

978030 
978§9§ 
978858 

•  67 

503982 

7-27 

496018 

18 

43 

483316 

6-59 

•  67 

5o44iS 

7-26 

495582 

17 

44 

4837  12 

6-58 

•  67 

5o4854 

7-25 

495146 

r6 

45 

484107. 

6-57 

978817 

•  67 

505289 

7-25 

4947 1 1 

i5 

46 

484501 

6-5- 

978777 

.67 

50372 i 

7-24 

494276 

i4 

47 

484895 

1   6-56 

978736 

-67 

5o6 1 59 

7-24 

493841 

i3 

48 

485289 

6-55 

1   978696 

-68 

5o65q3 

7-23 

493407  1  12  1 

49 

485682 

6-55 

978635 

-68 

507027 

7-22 

492973  1  II  1 

5o 

486075 

6-54 

978615 

1  -68 

507460 

:   7-22 

49-540 

10 

5i 

9-486467 

6-53 

9-978574 

•  68 

9-507893 

7-21 

10-492107 

0 

52 

4S6860 

6-53 

978333 

•  68 

5o832b 

7-21 

491674 

8 

53 

487251 

6-52 

078493 

•  68 

508759 

7-20 

491241 

7 

54 

487643 
48So34 

6.5i 

978432 

•  68 

509191 

7-19 

490809 

400378 

1   489946 

6 

55 

6-5i 

97841 1 

•  68 

509622 

]:\l 

5 

56 

48H404 

6-5o 

978370 

•  68 

5ioo54 

4 

u 

488S14 

6-5o 

9-8329 

97828S 

•  68 

510485 

7-i8 

489315 

3 

489204 

6-49 
6-48 

•  68 

510916 

7-17 

4890M4 
488'.54 

2 

59 

489503 
489982 

978247 

•  68 

5ii346 

7-l6 

I 

60 

6-48 

978206 

•  68 

511776 

7-16 

1   D. 

488224 

0 

Coaine 

D. 

1   Sine 

D. 

'  Cotane. 

1  Tats:— 

51. 

(72 

DEG 

REES.) 

86 


(18    DEGREES.)      A   TABLE   OF    LOGARITHMIC 


0 

Sine 

D. 

Cosine 

D. 

Tang. 

D.   1 

Cotang. 

9-489982 

6.48 

9-978206 

•  68 

9-511776 

7.16 

10-488224 

60 

I 

490371 

6-48 

978165 

•  68 

512206 

16 

487794 

5? 

i 

490759 

6-47 

978124 

•  68 

512635 

i5 

487365 

3 

491147 

6-46 

978083 

.69 

5i3o64 

14 

486q36 

57 

4 

491535 

6-46 

978042 

.69 

513493 

i4 

4865o7 

56 

5 

491922 

6-45 

978001 

•  69 

513921 
514349 

i3 

486079 

55 

6 

492308 

6-44 

977959 

•  69 

i3 

485651 

54 

I 

4926q5 

6-44 

977018 

.69 

514777 

12 

485223 

53 

493081 

6-43 

977877 

.69 

5i52o4 

12 

484796 

52 

9 

493466 

6-42 

977835 

.69 

5i563i 

II 

484369 

5i 

10 

493851 

6-42 

977794 

•  69 

5i6o57 

10 

483943 

5o 

II 

9-494236 

6-41 

9-977752 

.69 

9-516484 

10 

io-4835i6 

49 

13 

494621 

6-41 

9777>i 

.69 

516910 
517335 

09 

483090 

48 

i3 

493005 

6-40 

977669 

•  69 

09 

482665 

47 

14 

49^388 

6-39 

977628 

•  69 

517761 
5i8i85 

7 

08 

482239 

46 

i5 

495772 

6-39 

977586 

•69 

08 

48i8iD 

45 

i6 

496154 

6-38 

977544 

•70 

518610 

07 

481390 

44 

17 

496537 

6-37 

9775o3 

•  70 

519034 

06 

480966 
480342 

43 

i8 

496919 
497301 

6-37 

977461 

■  70 

519458 

06 

42 

«9 

6-36 

977419 

•  70 

519882 

o5 

480118 

41 

20 

497682 

6-36 

977377 

.70 

52o3o5 

03 

479695 

40 

21 

9-498064 

6-35 

9-977335 

.70 

9-520728 

04 

10-479272 

39 

22 

498444 

6-34 

977293 

•  70 

52ii5i 

03 

478849 

38 

23 

498825 

6-34 

977231 

.70 

521573 

o3 

478427 

37 

24 

490204 

6-33 

977209 

•  70 

521995 

o3 

478005 

36 

25 

499584 

6-32 

977167 
977125 

.70 

522417 
522838 

02 

477583 

35 

26 

499963 

6-35 

■  70 

02 

477 '62 

34 

11 

5oo342 

6-3i 

977083 

•  70 

523259 

01 

476741 

33 

500721 

6-31 

977041 

•  70 

523680 

01 

476320 

32 

29 

5o 1 099 

6-3o 

976999 

•  70 

524100 

I 

00 

475900 

3i 

So 

501476 

6-29 

976937 

•  70 

524520 

99 

475480 

3o 

3i 

9-5oi854 

6-29 

9-976914 

•  70 

9-524939 

52535o 

6 

99 

io-475o6i 

^"l 

32 

5o223i 

6.28 

976872 

•7« 

6 

98 

474641 

28 

33 

502607 

6.28 

97683o 

•71 

52577§ 

6 

93 

474222 

27 

34 

502984 
5o336o 

6-27 

976787 

•71 

526197 

6 

97 

4738o3 

26 

35 

6-26 

976745 

•7' 

526615 

6 

97 

473385 

25 

36 

5o3735 

6-26 

976702 

•71 

527033 

6 

96 

472967 
472549 

24 

ll 

5o4iio 

6-25 

976660 

•71 

527451 

6 

96 

25 

504485 

6-25 

976617 

•71 

527868 

6 

95 

472132 

22 

39 

5o486o 

6-24 

976574 

•  71 

528285 

6 

95 

471715 

21 

40 

5o5234 

6-23 

976532 

•71 

528702 

6 

94 

471298 

20 

41 

9 • 5o56o8 

6-23 

9-976489 

•71 

9-529119 
529535 

6 

93 

10-470881 

:? 

42 

505981 

6-22 

976446 

•  71 

6 

93 

470465 

43 

5o6354 

6-22 

976404 

•7> 

529950 

6 

93 

47oo5o 

«7 

44 

506727 

6-21 

976*61 

•71 

53o366 

6 

92 

469634 

16 

45 

507099 

6-20 

676318 

•71 

530781 

6 

9' 

469219 

i5 

46 

507471 

6-20 

976275 

•71 

531196 

6 

91 

468804 

14 

s 

507843 

6-19 

976232 

•72 

531611 

6 

90 

468389 
467975 
467561 

i3 

508214 

6- to 
6-18 

976189 

•72 

532025 

6 

2° 

12 

49 

5o8585 

976146 

•72 

532439 

6 

§9 

II 

5o 

508956 

6-18 

976103 

.72 

532853 

6 

89 

467147 

13 

5i 

9.509326 

6-17 
6-16 

9-976060 

•72 

9-533266 

6 

88 

10-466734 

I 

52 

509696 

976017 

•72 

533679 

6 

88 

466321 

53 

5ioo65 

6-16 

975974 

•  72 

534092 

6 

87 

465908 

I 

54 

510434 

6-i5 

975930 

•  72 

534304 

6 

87 

465496 

55 

5io8o3 

6-i5 

975887 

•72 

534916 

6 

86 

465o84 

5 

56 

511172 

6-14 

975844 

.72 

535328 

6 

86 

464672 
464261 

4 

s 

5ii54o 

6i3 

975800 

•72 

535739 

6 

85 

3 

511907 

6-i3 

975757 

•72 

536130 

6 

85 

463850 

a 

59 

512275 

6-12 

975714 

•72 

536561 

6 

84 

46343o 
463028 

I 

60 

512642 

6-12 

975670 

.72 

536972 

6-84 

0 

Cosine 

D. 

Sine   1 

D. 

Cotang. 

D. 

.Tang.  .,, 

M. 

(71    DEGREES.) 


SINES   AND   TANGENTS.      (19    DEGHEES.) 


37 


M. 

Sine 

1). 

Cosiuo 

D. 

'   Taiijj. 

D. 

Cotaug. 

0 

9-512642 

6-12 

9-975670 

.73 

9-536072 

537382 

6-84 

10-463028 

60 

I 

5i3ij09 

6-11 

973627 

•73 

6-83 

462618 

§ 

3 

5.337:) 

6-11 

975583 

.73 

53-7792 
538202 

6-83 

462208 

1 

3 

5i374i 

6-10 

975539 

•73 

6-82 

461798 

57 

< 

514107 

6  09 

975496 

•■7^ 

538611 

6-82 

461389 

56 

5 

514472 

6-09 
6-o8 

975432 

•73 

539020 

6-81 

460980 

55 

6 

5x4837 

975408 

.73 

539429 

6-8i 

460371 

54 

I 

5i5202 

6.08 

973365 

•73 

539837 

6-80 

460.63 

53 

5 15566 

6-07 

975321 

•73 

540245 

6-80 

439733 

52 

9 

5i5g3o 

6-07 

975277 

•73 

540653 

6-79 

439347 

5i 

• 

10 

516294 

6-06 

975233 

.73 

541061 

6-79 

458939 

5o 

II 

9.516657 

6-o5 

9.975189 

.73 

9.541468 

6-78 

10-458532 

S 

12 

517020 

6-o5 

975143 

•73 

541875 

6-78 

458125 

n 

517382 

6-04 

975101 

•73 

542281 

6-77 

457719 

47 

14 

517745 

6-04 

975057 

•73 

542688 

6.77 

4573.2 

46 

i5 

518107 

6-o3 

975oi3 

■73 

543og4 

6-76 

456906 

45 

i6 

518468 

6-03 

974969 
974925 

•74 

543499 

6-76 

456301 

44 

\l 

518829 

6-02 

•74 

54390J 
544310 

6-75 

456095 

43 

5191^0 

6-01 

974S80 

•74 

6-75 

455690 

42 

19 

SiqSdi 

6-01 

974836 

•74 

544715 

6-74 

455285 

41 

20 

519911 

6-00 

974792 

•74 

545119 

6.74 

454881 

40 

21 

9-520271 

6-00 

9.974748 

•74 

9.545524 

6-73 

10-454476 

39 

22 

52063 1 

5-99 

974703 

•74 

545928 
546331 

6-73 

454072 

38 

23 

520990 

5-99 

974659 

•74 

6-72 

453669 
453265 

37 

K 

24 

521349 

5-9§ 

974614 

•74 

546735 

6-72 

36 

1 

25 

521707 

5.98 

974370 

•74 

547138 

6-71 

452862 

35 

1 

26 

522066 

5-97 

974525 

•74 

547540 

6-71 

452460 

34 

w 

U 

522424 

5-96 

974481 

•74 

547943 

6-70 

452057 

33 

■■:■' 

522781 

5-96 

974436 

•74 

548345 

6-70 

45.655 

32 

, 

?9 

523i38 

5-95 

974391 

■74 

548747 

6-69 

45.253 

3i 

3o 

523495 

5-95 

974347 

•75 

549149 

6-69 

45o85i 

3o 

3i 

9-523852 

5-94 

9-974302 

•75 

9.549550 

6-63 

io-45o45o 

29 

32 

524208 

5-94 

974257 

•75 

54995 1 
55o332 

6-63 

45oo4q 

28 

33 

524564 

5-93 

974212 

•75 

6-67 

449648 

27 

34 

524920 

5-93 

974167 

•75 

55o752 

6-67 

449248 

26 

35 

525275 

5-92 

974122 

.75 

55ii52 

6-66 

448848 

25 

' 

36 

525630 

5-91 

974077 

•■'^ 

55i552 

6-66 

448448 

24 

37 
33 

525984 

5-91 

974032 

•75 

551952 

6-65 

448043 

23 

526339 
526693 

5-90 

973987 

•75 

552351 

6-65 

447649 

22 

39 

5-90 
5. §9 

973042 

973897 

•75 

552730 

6-65 

447250 

21 

40 

527046 

•75 

553149 

6-64 

446851 

20 

41 

9-527400 

5-8= 
5-89 

9-973852 

•75 

9.553548 

6-64 

10-446452 

19 

42 

527753 
528io5 

973807 

.75 

553946 
554344 

6-63 

446054 

18 

43 

5-83 

973761 

.75 

6-63 

445656 

\l 

44 

528458 

5.87 

973716 

.76 

554741 

6-62 

445259 

45 

528810 

5.87 

973671 

-76 

555i39 

6-62 

444861 

i5 

46 

529161 

5.86 

973625 

.76 

555536 

6.61 

444464 

14 

s 

529513 

5-86 

973580 

.76 

555933 
556329 
556725 

6-61 

444067 

i3 

529864 

5-85 

973535 

.76 

6-60 

443671 

13 

49 

53o2i5 

.  5-85 

973489 

.76 

6-60 

443275  ,  Ji  1 

bo 

53o565 

5-84 

973444 

.76 

557121 

6-59 

442879 

10 

5i 

9-530915 

5-84 

9-973398 

.76 

9-557517 

6-59 

10-442483 

0 

8 

5a 

531265 

5-83 

973332 

.76 

5579.3 

6-59 

442087 

53 

53i6i4 

5-82 

973307 

.76 

5583o8 

6-58 

441692 

I 

54 

531963 
532^12 

5-82 

973261 

.76 

558702 

6-58 

441298 

55 

5-81 

973215 

.76 

559097 

6-57 

44ooo3 
440309 
44011 5 

5 

56 

532661 

5-81 

973169 

.76 

559491 

6-37 

4 

u 

533009 

5-80 

973124 

•76 

559885 

6-56 

3 

533357 

5-80 

973078 

.76 

560279 
560673 

6-56 

439721 
439327 

3 

59 

533704 

^-'g 

973o32 

•77 

6-55 

I 

60 

534o52 

5.7§ 

972986 

•77 

56 I 066 

6-55 

438934 

0 

Cosine 

D. 

Sine 

D. 

Cotanj?. 

D. 

Tang. 

M. 

(70    DEGREES.) 


88 


(20   DEGREES.)     A  TABLE   OF  LOGARITHMIC 


M. 

Si  110 

J). 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9 -534052 

5-78 

9-972986 

•77 

9-56io66 

6.55 

10-43 8o34 

'  60 

I 

534399 

5-77 

972940 

•77 

561409 

6-54 

438341 

1? 

■> 

534743 

5-77 

972894 

•77 

1   56i85i 

6-54 

438149 

3 

5JJ092 

5-77 

972848 

•77 

562244 

6-53 

437736 

% 

!  ^ 

53 543 y 

5-76 

972802 

•77 

562636 

6-53 

437364 

^ 

1   535783 

5-76 

972755 

•77 

563028 

6-53 

436^72 

55 

1  6 

1   536i2g 

5-75 

972709 

•77 

563419 

6-52 

4363Si 

5i 

I 

536474 

5-74 

972663 

•77 

563811 

6-52 

436 i8q 

53 

1   536«i8 

5-74 

972617 

•77 

564202 

6-5i 

435798 

\   511 

9 

537163 

5-73 

972070 

•77 

564592 

6.5i 

435408 

5i 

10 

537507 

5-73 

972524 

•77 

564983 

6-5o 

435017 

5o 

II 

9-53785! 

5-72 

9-972478 

•77 

9-565373 

6-5o 

10-434627 

45 

48 

12 

538194 

5-72 

972431 

.78 

565763 

6-49 

434237 

i3 

538538 

5-71 

972385 

.78 

566 1 53 

6-49 

433847 

4-' 

14 

538880 

5.7. 

972338 

.78 

566542 

6.49 

433458 

46 

i5 

53g2  23 

5-70 

972291 

.78 

566932 

6-48 

433068 

45 

i6 

539565 

5-70 

972245 

.78 

567320 

6-48 

4326S0 

44 

«7 

539907 

5-69 

972198 

.78 

567709 

6-47 

432291 

43 

i8 

540249 

5-69 

972131 

.78 

568098 

6-47 

431902 

42 

19 

540090 

5-68 

972105 

•■^^ 

568486 

6-46 

43i3i4 

41 

20 

54093 1 

5-68 

972058 

.78 

568873 

6-46 

431127 

40 

21 

9-541272 

5-67 

9-972011 

.78 

9-569261 

6-45 

10-430739 

39 

38 

22 

54i6i3 

5.67 

97 1 964 

.78 

569648 

6-45 

43o332 

23 

541953 

5-66 

97i9'7 

.78 

570035 

6-45 

429965 

37 

24 

5422q3 

5-66 

971 870 

■^l 

570422 

6-44 

429078 

36 

25 

542632 

5-65 

971823 

.78 

570809 

6-44 

429191 

35 

26 

542971 

5-65 

971776 

.78 

571193 

6-43 

428803 

34 

27 

543310 

5-64 

971729 

•79 

571381 

6-43 

42S419 

33 

28 

543649 

5-64 

971682 

•79 

571967 

6-42 

428033 

32 

29 

543987 

5-63 

971635 

•79 

572352 

6-42 

427648 

3r 

3o 

544325 

5-63 

971588 

•79 

572738 

6-42 

427262 

3o 

3i 

9-544663 

5-62 

9-971540 

•79 

9-573i23 

6-41 

10-426877 

29 

28 

32 

545ooo 

5-62 

971493 

•79 

573307 

6-41 

426493 

33 

545338 

5-61 

971446 

•79 

573892 

6-40 

426108 

27 

34 

545674 

5-6i 

971398 

•79 

574276 

6-40 

425724 

26 

35 

5460 II 

5-60 

9713DI 

•79 

574660 

6-39 

425340 

25 

36 

546347 

5-60 

97i3o3 

•79 

575044 

6-39 

424956 

24 

H 

546683 

5-59 

971256 

•79 

575427 

6-39 

424073 

23 

3a 

547019 

5-59 

971208 

•79 

573810 

6-38 

424190 

32 

39 

547354 

5-58 

971161 

•79 

576193 

6-33 

423807 

21 

40 

547689 

5-58 

971113 

•79 

576576 

6.37 

423424 

20 

41 

9.548024 

5-57 

9-971066 

.80 

9-576938 

6-37 

io-423o4i 

IQ 

42 

548359 

5.57 

971018 

.80 

577-^41 

6-36 

422659 

18 

i^T. 

548693 

5-56 

970970 

.80 

577723 

6-36 

422277 

n 

44 

549027 

5-56 

970922 

.80 

578104 

6-36 

421896 

16 

45 

549360 

5-55 

970874 

.80 

5784% 

6-35 

42i5i4 

i5 

46 

549693 

5-55 

970S27 

.80 

578067 
579248 

6-35 

421133 

14 

47 

550026 

5-54 

970779 

.80 

6-34 

420732 

i3 

48 

55o359 

5-54 

970731 

-80 

579629 

6-34 

420371 

12 

49 

550692 

5-53 

970083 

•  80 

580009 

6-34 

4IV991 

II 

5o 

55io24 

5-53 

970635 

.80 

58o389 

6-33 

419611 

10 

5i 

9-55i356 

5-52 

9-970586 

•  80 

9-580769 

6-33 

10-419231 

I 

52 

551687 

5-52 

970538 

.80 

581149 

6-32 

4i885i 

53 

552018 

5-52 

970490 

.80 

58i528 

6-32 

418472 

7 

54 

552349 

5.5: 

970442 

.80 

581907 

6-32 

41S093 

6 

55 

552680 

5-51 

970394 

.80 

582  286 

6-3i 

417714 

5 

56 

553oio 

5-5o 

970345 

•  81 

582665 

6-3i 

417335 

4 

% 

553341 

5-5o 

970297 

•  81 

583043 

6-3o 

416937 

3 

553670 

5-49 

970249 

-81 

583422 

6-3o 

416378 

2 

5q 

554000 

5-49 

970200 

•  81 

583800 

6-29 

4105OO 

I 

66 

554329 

5-48 

970i52 

•81 

584177 

6-29 

415823 

0 
M. 

Coeine 

D.   1 

Sine 

D. 

Cotanpr. 

D. 

Tang. 

(69  DSaREES.) 


SIXES  AND  TAXGEXT3.      (21    DEGREES.) 


39 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

1). 

Coting. 

60 

o 

9-554329 

5-43 

9-970152 

.81 

9-584177 

6-29 

!0-4i5823 

I 

554')5S 

5-48 

970103 

•  81 

584555 

6-29 

415445 

U 

a 

5549"^7 

5-47 

970055 

.81 

584932 

6-28 

41306S 

3 

5553 1 5 

5-47 

970006 

-81 

585309 

6-28 

414691 

57 

4 

555643 

5-46 

969957 

•  81 

585686 

6-27 

414314 

56 

5 

555971 

5.46 

969909 
969860 

•  81 

•586062 

6-27 

413938 

55 

6 

556299 

5-45 

.81 

5S6439 

6-27 

413361 

54 

7 

556626 

5-45 

969811 

•  81 

58681 D 

6-26 

4i3i85 

53 

1  8 

556953 

5-44 

969762 

•  81 

5S7190 

6-26 

412S10 

32 

9 

557280 

5-44 

969714 

•  81 

587366 

6-25 

412434 

5l 

10 

557606 

5-43 

969665 

.81 

587941 

6-25 

412039  1  5o  1 

II 

g-55i932 

5-43 

9 -9696 1 6 

.82 

9-5883i6 

6-25 

10-411684 

49 

12 

5582 i8 

5-43 

969567 

•  82 

588691 

6-24 

41 i3o9 

48 

i3 

558583 

5-42 

969318 

•82 

589066 

6-24 

410934 

47 

U 

558909 

5-42 

969469 

■  82 

5S9440 

6-23 

4 1 0360 

46 

i5 

559204 

5-41 

969420 

-82 

589814 

6-23 

410186 

45 

i6 

559558 

5-41 

969370 

•82 

5qoi88 

6-23 

409S1 2 

44 

\l 

559883 

5-40 

969321 

•82 

590562 

6-22 

409438 

43 

560207 

5-40 

969272 

-82 

590935 

6-22 

400065 
408692 

42 

»9 

56o53 I 

5-39 

969223 

•  82 

591308 

6-22 

41 

20 

56o855 

5-39 

969173 

.82 

591681 

6-21 

4083 1 9 

40 

21 

9.56117S 

5.33 

9-969124 

.82 

^•592054 

6-21 

10-4079^6 

39 

22 

56i5oi 

5-33 

969075 

.82 

592426 

6.20 

407374 

38 

23 

561824 

5.37 

969025 

.82 

592798 

6-20 

407202 

37 

24 

562146 

5.37 

968976 

.82 

593170 

6-19 

406829 

36 

25 

562468 

5-36 

968926 

.83 

593542 

6-19 

406438 

35 

26 

562790 

5-36 

968877 

•  83 

593914 

6-18 

406086 

34 

11 

563 1 1 2 

5-36 

968827 

.83 

594285 

6-18 

40571 5 

33 

563433 

5-35 

968777 
968728 

•  83 

5g4656 

6-18 

405344 

32 

?9 

563:55 

5-35 

■  83 

595027 

6-17 

404973 

3i 

3o 

564075 

5-34 

968678 

•  83 

595398 

6-17 

404602 

3o 

3i 

9-564396 

5-34 

9-968628 

•  83 

9-595768 

6-17 

6-16 

10-404232 

29 

32 

564716 

5-33 

068578 

•  83 

596133 

4o3S62 

28 

33 

065o36 

5.23 

968528 

•83 

596508 

6^i6 

403492 

27 

34 

5v  5356 

5.32 

968479 

.83 

596878 

6^i6 

4o3i22 

26 

33 

5(  5676 

5-32 

968429 

•  83 

597247 

6^i5 

402753 

25 

36 

565995 
566314 

5-3i 

968379 

•  83 

597616 

6-i5 

402384 

24 

h 

5-3i 

968329 

.83 

597985 
:'9S354 

6^i5 

40201 5 

23 

38 

56'j632 

5-3i 

968278 

•  83 

6-14 

401646 

22 

39 

566951 

5-3o 

968228 

•84 

598722 

6^i4 

401278 

21 

40 

567269 

5-3o 

968178 

•84 

599091 

6^13 

400909 

20 

41 

9-567587 

5-29 

9-968128 

.84 

9-599459 

6^i3 

10 -40054 1 

19 

42 

567904 

5-29 

968078 

.84 

599827 

6.i3 

400173 

18 

43 

5682  2  2 

5-28 

968027 

.84 

600194 

6-12 

399806 

\l 

44 

568539 

5-23 

967977 

.84 

6oo562 

6-12 

399438 

45 

568856 

5-28 

967927 

.84 

600929 

6-11 

399071 

i3 

46 

569172 

5-27 

967870 

.84 

601296 

6-11 

39S704 

14 

s 

569488 

5-27 

967826 

•84 

601662 

6-11 

398338 

i3 

569804 

5-26 

967775 

.84 

612029 

6-10 

397971 

12 

^9 

570120 

5-26 

967725 

.84 

6t>2393 

6-10 

397605 

11 

5o 

570435 

5-25 

Q67674 

.84 

602761 

6^10 

397239 

10 

5i 

9-570751 

5-25 

9-967624 

.84 

9-6o3i27 

6-09 

10  396873 

9 

57 

571066 

5-24 

967573 

.84 

603493 

6-09 

396307 

8 

53 

571380 

5-24 

967522 

.85 

6o3858 

6.09 

396:42 

7 

04 

571695 

5-23 

967471 

.85 

604223 

6-08 

3q5777 

6 

55 

572009 

5-23 

967421 

.85 

604388 

6-o3 

395412 

5 

56 

572323 

5-23 

967370 

•  85 

604953 
6o53i7 

6-07 

395047 

4 

5- 

572636 

5-22 

967319 

•  85 

6-07 

394683 

3 

58 

572920 

5-22 

967268 

•  85 

603682 

6-07 

394318 

2 

59 

573263 

5-21 

9672:7 

•  85 

606046 

6-06 

393954 

I 

6fj 

573575 

5-21 

967166 

•  85 

606410 

6-o6 

393390 

0 

1 

Cosine 

I   D. 

Sine 

1  D. 

'  Cotan?. 

D. 

Tang. 

M. 

(68  DEGREES.) 


40 


(22   DEGREES.)     A  TABLE   OF   L0C5AEITHMIC 


M. 

0 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

9  573575 

5.21 

9-967166 

.85 

9-606410 

6«o6 

10-393593 

to 

I 

5738^8 

5-20 

9671 1 5 

.85 

606773 

6-o6 

393227 

il 

2 

574200 

5-20 

967064 

.85 

607137 

6-o5 

392863 

3 

574513 

5-19 

967013 

.85 

6o-i5oo 

6-o5 

392500 

U 

4 

574834 

5.19 

966961 

.85 

607863 

6-64 

392137 

5  1 

5751 36 

5-19 

9669 1 0 

•  85 

608225 

6-04 

391775 

55 

6  i 

575447 
57575^ 

5-18 

966839 

.85 

6o8588 

6-04 

391412 

54 

i! 

5.18 

966808 

.85 

608950 
6093 1 2 

6-o3 

391050 

53 

576069 

5-17 

966756 

.86 

6-o3 

390688 

52 

9 

576379 

5-17 

966705 

.86 

609674 

6-03 

390326 

5i 

to 

576689 

5-16 

966653 

.86 

6ioo36 

6-02 

389964 

5o 

II 

9-576099 
577309 

5-16 

9  966602 

.86 

9-610397 
610739 

6-02 

10-389603 

49 

13 

5-16 

966550 

-86 

6-02 

389241 

48 

i3 

577618 

5-15 

966499 

-86 

611120 

6-01 

388880 

^I 

14 

577937 

5-i5 

966447 

.86 

611480 

6-01 

388520 

45 

i5 

578336 

5-14 

966395 

.86 

611841 

6-01 

388:59 

45 

i6 

578545 

5-14 

966344 

.86 

612201 

6-00 

387709 
387439 

44 

17 

578853 

5-i3 

966292 

•  86 

6i256i 

6-00 

43 

i8 

579163 

5-i3 

966240 

.86 

612921 

6-00 

387079 

42 

'9 

579470 

5-i3 

9661S8 

.86 

6132S1 

5-99 

386719 

41 

JO 

579777 

5-12 

966136 

.86 

6i364i 

5-99 

386359 

40 

31 

9-5Soo85 

5-12 

9-966085 

.87 

9-614000 

5-98 

io-386ooo 

39 

23 

580392 

5-n 

966033 

-87 

614359 

^98 

385641 

38 

33 

580699 
58ioo5 

5. II 

965981 

.87 

614718 

5-98 

385282 

37 

34 

5-II 

965928 

.87 

61 5077 

5-97 

384923 

36 

35 

58i3i3 

5-10 

965S76 

•87 

615435 

5-97 

384565 

35 

36 

58i6i8 

5-10 

965824 

•87 

615793 
6i6i5i 

5-97 

384207 

34 

3 

581924 

5-09 

965772 

•87 

5-96 

383849 

33 

582229 
582535 

5-09 

965720 

•87 

6i65o9 

5.96 

383491 

32 

39 

5-09 

965668 

•  87 

616867 

^96 

383 1 33 

3i 

3o 

582840 

5.o§ 

9656 1 5 

.87 

617224 

5-95 

3S2776 

3o 

3i 

9-583145 

5-08 

9-965563 

.87 

9-617582 

^95 

10-382418 

2 

33 

583449 

5-07 

965511 

.87 

617939 
618295 

5-95 

382061 

33 

583754 

5.07 

965458 

.87 

5.94 

381705 

?7 

34 

584o58 

5-06 

965406 

.87 

6i8652 

5-94 

38i348 

26 

35 

584361 

5.06 

965353 

.88 

619008 

5-94 

380993 
38o636 

25 

36 

584665 

5-06 

965301 

.88 

619364 

5.93 

24 

u 

684968 

5-o5 

965248 

-88 

619721 

^93 

380279 

23 

585272 

5-o5 

965195 

.88 

620076 

5-93 

379924 

23 

39 

5S5574 

5-04 

965143 

.88 

620432 

5-92 

379568 

21 

40 

585877 

5-04 

965090 

.88 

620787 

5-92 

379213 

20 

41 

9-586179 

5-o3 

9-965o3i 

.88 

9-621142 

5-92 

ic-378858 

'2 

42 

586482 

5-o3 

9649S4 

.88 

621497 

5-91 

3785o3 

18 

43 

586783 

5-o3 

964931 

.88 

621802 

5-91 

378148 

17 

44 

587085 

5-02 

964879 

.88 

622207 

5-90 

377793 

16 

45 

587386 

5-02 

964826 

.88 

622361 

5.90 

377439 
377085 

i5 

46 

587688 

5-01 

964773 

.88 

622915 

5-90 

i4 

S 

587989 

5-01 

964719 

.88 

623269 
623623 

5-§9 

376731 

i3 

588289 

5-01 

964666 

.89 

5-89 

376377 

12 

49 

588590 

5-00 

964613 

.89 

623976 

5-89 

376024 

IF 

5o 

588890 

5-00 

964560 

.89 

624330 

5-88 

375670  !  13 

i 

5i 

9-589190 

589489 

4-99 

9.964507 

.89 

9-624683 

5-88 

10.375317     9 

5j 

4-99 

964454 

.89 

625o36 

5-88 

374964   8 

53 

589789 
5goo88 

4-99 

964400 

-89 

625388 

5-87 

374612 

I 

54 

4-98 

964347 

-89 

625741 

5-87 

374259 

55 

590387 

4.98 

964294 

-89 

626093 

5-87 

570907 
373355 

5 

56 

590686 

4-97 

964240 

-89 

626445 

5-86 

4 

u 

590984 

4-97 

964187 

-89 

626797 

5-86 

373203 

3 

(   591282 

4-97 

964133 

-89 

627149 

5-86 

372851 

3 

59 

1   5oi58o 

4-96 

96408c 

-89 

627501 

,   5-85 

372199 

372I4t» 

I 

66 

591878 

4-96 

964026 

.89 

627S52 

1   5-85 

1 

c 

Ocsme 

D. 

Sine 

D. 

Cotang. 

1   D. 

Tai«. 

n. 

(67 

DEOl 

^EES.) 

SINKS  AND  TANGENTS 

.   (23  DEGREES.) 

41 

M.  1 

0 

Sine 

D. 

Cosine 

D.  1 

Tanu. 

D. 

Cotang. 

9.591878 

4.96 

9-964026 

.89 

9-627852 

5-85 

10-372148 

60 

I 

592176 

4-95 

963972 

.89 

628203 

5-85 

371797 

59 

1 

592473 

4-95 

963919 

.89 

628554 

5-85 

371446 

58 

3 

592770 

4-95 

963863 

-90 

628905 

5-84 

371095 

a 

4 

5g3o67 

4-94 

96381 1 

-90 

629235 

5-84 

370743 

5 

593363 

4-94 

963757 

.90 

629606 

5-83 

370394 

55 

6 

593639 

4-93 

963704 

.90 

629956 

5-83 

370044 

54 

I 

593953 

4-93 

96365o 

.90 

63o3o6 

5-83 

369694 

53 

594251 

4-93 

963596 

-90 

63o656 

5-83 

369344 

52 

9 

594547 

4-92 

963542 

.90 

63ioo5 

5-82 

368995 

5i 

10 

594842 

4-92 

963488 

.90 

63i355 

5-82 

368643 

5o 

II 

9-595i37 

4-91 

9-963434 

.90 

9-631704 

5-82 

10-368296 

4q 

12 

595432 

4-91 

963379 

.90 

632053 

5-81 

367947 
367599 

48 

i3 

595727 

4-91 

96332  3 

.90 

632401 

5-8i 

47 

14 

596021 

4-90 

963271 

.90 

632730 

5-81 

367230 

46 

i5 

5963 1 5 

4-90 

963217 

.90 

633098 

5 -80 

366902 

45 

i6 

596609 
596903 

4-89 

9631 63 

.90 

633447 

5-80 

366353 

44 

\l 

4-89 

963108 

•  91 

633795 

5-80 

3662o5 

43 

597196 

4-89 

963o54 

.91 

634143 

5-79 

365857 

42 

19 

597490 

4-88 

962999 
962945 

.91 

634490 

5-79 

365510 

41 

20 

597783 

4-88 

-91 

634838 

5-79 

365i62 

40 

31 

0.598075 

4-87 

9-962890 
962836 

•91 

9-635i85 

5-78 

io-3648i5 

39 

38 

32 

598368 

4-87 

-91 

635532 

5-78 

364468 

a3 

598660 

4-87 

962781 

.91 

635879 

5-78 

364121 

37 

24 

598952 

4-86 

962727 

-91 

636226 

5-77 

363774 

36 

25 

599244 

4-86 

962672 

•91 

636572 

5-77 

363428 

35 

26 

599536 

4-85 

962617 

.91 

636919 
637265 

5-77 

363o8i 

34 

11 

59982-7 
600118 

4-85 

962362 

-91 

5-77 

362735 

33 

4-85 

962508 

.91 

637611 

5-76 

362389 

I' 

29 

600409 

4-84 

962453 

•91 

637936 
6383o2 

5-76 

362044 

3i 

3o 

600700 

4-84 

962398 

•  92 

5-76 

361698 

3o 

3i 

9-600990 

4-84 

9-962343 

-92 

9-638647 

5-75 

io-36i353 

2Q 

32 

601280 

4-83 

962288 

.92 

638992 
639337 

5-75 

361008 

28 

33 

601570 

4-83 

962233 

•  92 

5-75 

36o663 

27 

34 

601860 

4-82 

962178 

-92 

639682 

5-74 

36o3i8 

26 

35 

602 i5o 

4-82 

962123 

.92 

640027 

5-74 

359973 

25 

36 

602439 
602728 

4-82 

962067 

•  92 

640371 

5-74 

359629 

24 

u 

4-81 

962012 

-92 

6407 1 6 

5-73 

359284 

23 

6o3oi7 

4-8i 

961957 

.92 

641060 

5-73 

35.8940 
358596 

22 

39 

6o33o5 

4-8i 

961902 

.92 

641404 

5-73 

31 

40 

603594 

4-8o 

961846 

.92 

641747 

5-72 

358253 

20 

41 

9 -603882 

4-8o 

9-961701 
961735 

•  92 

9-642001 
642434 

5-72 

10-357909 

357566 

10 

42 

604170 

4-79 

•  92 

5-72 

18 

43 

604437 

4-79 

961680 

-92 

642777 

5-72 

357223 

\l 

44 

604743 

4-79 

961624 

-93 

643120 

5-71 

356880 

45 

6o5o32 

4-78 

961369 
96i5i3 

•93 

643463 

5-71 

356537 

i5 

46 

6o53i9 

4-78 

•93 

643806 

5-7« 

356194 

14 

47 

6o56o6 

4-78 

961458 

•93 

644148 

5-70 

355832 

i3 

48 

605892 

4-77 

961402 

.93 

644490 

5-70 

355510 

12 

49 

606179 

4-77 

961346 

•93 

644832 

5-70 

355i68 

II 

60 

606465 

4-76 

961290 

•93 

645174 

5-69 

354826 

10 

-51 

9  606731 

4-76 

9-961235 

.93 

9-645516 

5-69 

10-354484 

I 

ii 

607036 

4-76 

961179 
961123 

-93 

643857 

5-69 

354143 

53 

607322 

4-75 

-93 

646199 

5-69 
5-68 

353801 

I 

54 

607607 

4-75 

96 1 067 

•93 

646540 

353460 

55 

607892 
608177 

4-74 

961011 

.93 

646881 

5-68 

353119 

352778 

5 

56 

4-74 

960955 
960899 
960843 

•93 

647222 

5-68 

4 

u 

608461 

4-74 

•93 

647562 

5-67 

352438 

3 

608745 

4-73 

-94 

647903 
64^243 

5-67 

352097 
351757 

3 

59 

609029 
6093 1 3 

4-73 

960786 

.94 

5-67 

I 

60 

4-73 

960730 

.94 

648583 

5-66 

1 

351417 

0 

'  Cosine 

1   D. 

Sine 

1  D. 

'  Cotan^. 

1   D. 

i  Tonff. 

M. 

(66    DEGREES.) 


i 


42 


(24   DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

0 

Sine 

»•  ; 

O'/sine 

D. 

Tang. 

1>. 

Cotang. 

! 
60 

9-6o93i3 

4-73 

9-960730 

.94 

9-648583 

5-60 

10-335417 

I 

609397 

4-72 

960674 

•94 

648923 

5-66 

351077 

5^ 

3 

609880 

4-72 

960G18 

•94 

649263 

5-66 

350737 
350)98 

3 

610164 

4-72 

960361 

•94 

649')02 

5-66 

u 

4 

610447 

4-71 

96o5o5 

•94 

649942 

5-65 

35oo38 

5 

610729 

4-71 

960448 

.94 

65o28i 

5-65 

349719 

55 

6 

61 1012 

4-7^ 

960392 

•94 

630620 

5-65 

349380 

54 

I 

61 1294 

4-70 

960335 

•94 

650939 

5-64 

349041 

53 

611376 

4-70 

960279 

-94 

65i297 

5-64 

3487.-3   52  1 

9 

6ii858 

4.69 

960222 

•94 

631636 

5-64 

348364   ' 

5i 

10 

612140 

4.69 

960165 

•94 

651974 

5-63 

348026 

5n 

II 

9'6i242i 

4.69 
4-68 

9-960109 

.95 

9-652312 

5-63 

10-347688 

49 

12 

612702 

960032 

.95 

652630 

5-63 

347350   48 

i3 

612983 

4-68 

939995 

.95 

652988 

5-63 

347012   47 

14 

6i3264 

4-67 

939938 

.95 

653326 

5-62 

346674 

46 

i5 

613343 

4-67 

939882 

.95 

653663 

5.62 

346337 

45 

i6  1 

6i3823 

4-67 

939S25 

.93 

654000 

5-62 

346000 

44 

17  i 

6i4io5 

4-66 

939768 

.95 

654337 

5.61 

345663 

43 

i8 

614385 

4-6d 

9397 1 1 

.93 

654674 

5-61 

345326 

42 

•9 

614665 

4-66 

959634 

•  93 

655oii 

5-6i 

3449^9 

41 

20 

614944 

4-65 

939596 

.95 

655348 

5.61 

344632 

40 

21 

9-613223 

4-65 

9-939539 

.95 

9-655684 

5-60 

10-3443 16 

h 

22 

613302 

4-65 

939482 

.95 

656020 

5-60 

343980 

38 

23 

613731 

4-64 

939425 

.93 

656356 

5-6o 

343644 

ll 

24 

6 1 6060 

4-64 

939368 

.95 

636692 

5-59 

3433o8 

25 

61 6338 

4-64 

959310 

.96 

657028 

5-39 

342972 

3; 

36 

616616 

4-63 

939233 

.96 

657364 

5-59 

342636 

34 

U 

616894 

4-63 

939195 

.96 

657699 
658oJ4 

5.59 

342301 

33 

617172 

4-62 

959138 

.96 

5-58 

341966 

33 

39 

617430 

4-62 

959081 

.96 

658369 

5-58 

34i63i 

3i 

3o 

617727 

4-62 

959023 

.96 

65S704 

5-58 

341296 

So 

li 

9' 6 18004 

4-61 

9  058965 

.96 

9-659039 

5-58 

10-340961 

29 
28 

J2 

618281 

4-6i 

958908 
958850 

.96 

639373 

5-57 

340627 

J3 

618338 

4-61 

.96 

659708 

5-57 

340292 
339958 

U 

J4 

618834 

4-60 

938792 

.96 

660042 

5-57 

}5 

6191 10 

4-60 

938734 

.96 

660376 

5.57 

3396^4 

25 

56 

619J86 

4-6o 

938677 

.96 

6607 1 0 

5-56 

339290 
3389''7 

24 

\l 

619662 

4-59 

958619 

.96 

661043 

5-56 

23 

619938 

4-59 

958561 

.96 

661377 

5.56 

338623 

2? 

h 

620213 

4-59 

9585o3 

•97 

66 1 7 1 0 

5.55 

338290 

21 

40 

620488 

4-58 

958445 

•97 

662043 

5-55 

337937 

JO 

41 

9-620763 

4-58 

9-958387 

•97 

9-662376 

5-55 

10-337624 

19 

42 

621008 

4-57 

958329 

•97 

662709 

5-54 

33729. 

18 

43 

62i3i3 

4-57 

958271 

•97 

663o42 

5-54 

336Q38 

n 

44 

621387 

4-57 

938213 

•97 

663375 

5-54 

336625 

16 

4 -J 

621861 

4-56 

9581 54 

•97 

663707 

5-54 

3362g3 

i5 

46 

622135 

4-56 

958096 

•97 

664039 

5-53 

333961 

14 

47 

622409 

4-56 

95So38 

•97 

664371 

5-53 

335629 

i3 

48 

622682 

4-55 

9''7979 

•97 

664703 

5-53 

333297 

12 

49 

022956 

4-55 

957921 

•97 

663035 

5-53 

334963 

I' 

5o 

623229 

4-55 

957863 

•97 

665366 

5-52 

334634 

10 

ai 

9-623302 

4-54 

9-957804 

•97 

9 -665697 

5-52 

io-3:43o3 

I 

C/J 

623774 

4-54 

937746 

.98 

666029 

5-52 

333ij7i 

03 

624047 

4-54 

957687 

.98 

666360 

5-5i 

333640 

7 

54 

624319 

4-53 

931628 

.98 

666691 

5-5i 

333309 

6 

55 

624391 

4-53 

957570 

.98 

667021 

5-51 

332979 

5 

56 

624863 

4-53 

957511 

.98 

667352 

5-5i 

332648 

4 

U 

625i35 

4-52 

937452 

.98 

667682 

5-5o 

3323i8 

3 

623406 

4-52 

957393 
95-'335 

.98 

66801 3 

5-50 

33.987 

3 

.59 

625677 

4-52 

.93 

668343 

5-50 

33i657 

I 

66 

62594S 

4'5i 

957276 

.98 

668672 

5-5o 

33i328 

0 
M. 

Coeiiie 

D. 

Sine 

1  D. 

Cotansj. 

D. 

1   Tang. 

(65  DEGREES.) 


SINES   AND  T^ySTGENTS.      (25    DEGREES.) 


43 


0 

Sine 

D. 

Cosine 

D.  1 

Tang. 

D. 

Cotang. 

9-625948 

4-5i 

9.957276 

.93 

9-668673 

5.50 

ro-33i327 

330998 

60 

I 

626219 

4-5i 

937217 

.98 

669002 

5.49 

U 

3 

626490 

4-5i 

957153 

.93 

669332 

5.49 

330668 

3 

626760 

4-5o 

937099 

.98 

6(3966 1 

5-49 

33o33g 

5? 

4 

627030 

4 -50 

957040 

.93 

66999 1 

5-48 

33ooog 

56 

5 

627300 

4-5o 

956981 

.98 

670320 

5-48 

329680 

55 

6 

627370 

4.49 

956921 

•99 

670649 

5-48 

329351 

54 

I 

627840 

4.49 

956862 

•99 

670971 

5-43 

329023 
328694 

53 

628 [09 

4-49 

9568o3 

•99 

671306 

5-47 

5s 

9 

628378 

4.48 

956744 

•99 

671634 

5-47 

328366 

5i 

10 

628647 

4-48 

956684 

•99 

671963 

5-47 

328037 

5o 

II 

9  528916 

4-47 

9.956625 

•99 

9-672291 

5-47 

10.327709 

^? 

IS 

62^185 

4-47 

956566 

■99 

672619 

5.46 

327381 

t1 

6-'9453 

4-47 

9565o6 

•99 

672917 

5.46 

327053 

47 

U 

629j')i 

4-46 

956447 

•99 

673274 

5-46 

326726 

46 

i5 

6299^9 

4-46 

956387 

•99 

673602 

5.46 

326398 

45 

i6 

63025; 

4-46 

956327 
956268 

•99 

673929 

5.45 

326071 

44 

17 

63o524 

4.46 

•99 

674257 

5-45 

325743 

43 

i8 

630792 

4-45 

956208 

I -00 

674584 

5-45 

325416 

42 

19 

63 1 039 

4-45 

956143 

I -00 

674910 

5-44 

325090 

41 

20 

63i326 

4-45 

956089 

I -00 

675237 

5-44 

324763 

40 

21 

9- 63 1 593 

4-44 

9.956029 

I -00 

9.675564 

5-44 

10.324436 

39 

22 

63 1859 

4.44 

955969 

I -00 

675890 

5-44 

324110 

33 

23 

63212D 

4-44 

955909 
955849 

I-OO 

676216 

5-43 

323784 

37 

24 

632392 

4-43 

1. 00 

676543 

5-43 

323457 

36 

23 

632658 

4-43 

955789 

1. 00 

676869 

5-43 

323 i3 I 

35 

26 

632923 

4.43 

955729 

I. CO 

677194 

5.43 

322806 

34 

27 

633189 

4-42 

955669 

I-OO 

677520 

5.42 

322480 

33 

23 

633454 

4-42 

955609 

I-OO 

67-846 
67S171 

5-42 

322i54 

32 

29 

633719 

4-42 

955548 

1. 00 

5-42 

321829 

3i 

3o 

6339S4 

4-41 

955488 

1. 00 

678496 

5-42 

32i5o4 

3o 

3i 

9-634249 

4-41 

9.955423 

1. 01 

9-678821 

5-41 

10-321179 

29 
28 

32 

6345.4 

4.40 

955368 

1. 01 

679146 

5-41 

320854 

33 

634773 

4-40 

955307 

I-OI 

679471 

5-41 

320529 

27 

34 

635042 

4-40 

955247 

I. 01 

679795 

5-41 

32o2o5 

26 

35 

6353o6 

4-39 

955186 

I-OI 

6S0120 

5-40 

319880 

25 

36 

635570 

4-39 

955126 

1.01 

680444 

5-40 

319556 

24 

37 

635834 

4-3o 

g35o65 

1. 01 

680768 

5-40 

310232 

23 

38 

636097 

4-3S 

955oo5 

I-OI 

681092 

5-40 

318908 

22 

39 

636360 

4-33 

954944 

I-OI 

681416 

5.39 

3 1 8584 

21 

40 

636623 

4-33 

954883 

l-OI 

681740 

5.39 

318260 

20 

41 

9-636886 

4-37 

9.954823 

I-OI 

9.682063 

5.39 

10-317937 

19 

42 

637148 

4-37 

954762 

I-Ol 

682387 

5-39 

317613 

18 

43 

637411 

4-37 

954701 

I-Ol 

682710 

5-38 

317290 

'7 

44 

637673 

4-37 

954640 

I-Ol 

683o33 

5-38 

316967 

16 

45 

637935 

4-36 

954579 

I-OI 

683356 

5-38 

316644 

i5 

46 

638197 

4-36 

954518 

1-02 

683679 

5.38 

3 1632  1 

14 

47 

638458 

4-36 

954457 

1.02 

684001 

5.37 

3 1 5999 

i3 

48 

638720 

4-35 

954396 

1.02 

684324 

5-37 

315676 

12 

4g 

638981 

4-35 

954335 

1-02 

684646 

5.37 

313354 

11 

5o 

639242 

4-35 

934274 

1-02 

684968 

5-37 

3i5o32 

10 

5i 

9-6395o3 

4-34 

g-954213 

102 

9.685290 

5-36 

io-3i47io 

I 

52 

639764 

4-34 

954152 

I  02 

685612 

5.36 

3i4388 

53 

640024 

4-34 

954090 

1-02 

685934 

5.36 

314066 

7 

54 

640284 

4-33 

954029 

1-02 

686255 

5.36 

3i3745 

6 

55 

640544 

4 -33 

953968 

102 

686577 

5-35 

3i3423 

5 

56 

640804 

4-33 

95I906 

1-02 

686898 

5-35 

3i3i02 

4 

u 

641064 

<-32 

953843 

1  .02 

687219 

5-35 

312781 

3 

641324 

i-32 

933783 

1-02 

687540 

5-35 

312460 

J 

59 

641 584 

1-32 

953722 

I -03 

687861 

5-34 

312-39 

I 

60 

64184} 

4-3i 

953660 

1.(3 

688182 

5.34 

3ii»i8 

0 

Cobine 

D. 

Sine 

1  D. 

Cotati?. 

D. 

1   Tauir.  1  M. 

(6-1 

DEGR 

EES.) 

lA 


(26   DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

Sine 

D. 

Cosine    D 

^ 

Tang. 

D. 

Cotang. 

0 

9-641842 

4-3i 

9-953660   I- 

o3 

9-688182 

5-34 

io-3ii8i8 

60 

I 

642101 

4- 

3i 

953599   I  • 
953537   I- 

o3 

688502 

5- 

34 

311498   5q  1 

3 

642360 

4 

3i 

o3 

688823 

5 

34 

311177 

58 

3 

642618 

4 

3o 

953475   I  • 

o3 

689143 

5 

33 

310837 

57 

4 

642877 

4 

3o 

953413   J- 

o3 

689463 

5 

33 

3io537 

56 

5 

643 1 35 

4 

3o 

953352   I  • 

o3 

689783 

5 

33 

3ic2i7 

55 

6 

643393 

4 

3o 

953290   1 

o3 

6goio3 

5 

33 

309897 

54 

7 

6436DC 

4- 

29 

953228   I- 

o3 

690423 

5 

33 

309577 

53 

8 

643908 

4 

29 

953166   I 

o3 

690742 

5 

32 

309258 
308938 

52 

9 

644165 

4 

29 

953104   I 

o3 

691062 

5 

32 

5i 

10 

644423 

4 

28 

953042   I  • 

o3 

691381 

5 

32 

308619 

5o 

II 

9-644680 

4 

28 

9-952980   I 

04 

9-691700 

5 

3i 

io-3o83oo 

S 

12 

644936 

4 

28 

952918   I- 

04 

692019 

5 

3i 

307981 

i3 

645193 

4 

27 

952855   I 

04 

692338 

5 

3i 

307662 

47 

14 

645450 

4 

27 

952793   1 
9527J1    I 

04 

692656 

5 

3i 

307344 

46 

i5 

645706 

4 

27 

04 

692975 

5 

3i 

307025 

45 

i6 

643962 

4 

26 

952669   I 

04 

693293 

5 

3o 

306707 

44 

\l 

646218 

4 

26 

952606   I 

04 

693612 

5 

3o 

3o6388 

43 

646474 

4 

26 

952544   1 

04 

693930 

5 

3o 

306070 

42 

'9 

646729 

4 

25 

952481   I 

04 

694248 

5 

3o 

3o5752 

41 

20 

646984 

4 

25 

952419   I 

04 

694566 

5 

29 

3o5434 

40 

21 

9-647240 

4 

25 

9-952356   I 

04 

9-69^883 

5 

29 

ic-3o5ii7 

39 

23 

647494 

4 

24 

952294   I 

04 

695201 

5 

29 

3o4799 

38 

23 

647749 
648004 

4 

24 

952231   I 

04 

695518 

5 

29 

304482 

37 

24 

4 

24 

952168   I 

o5 

695836 

5 

29 

304164 

36 

25 

648258 

4 

24 

952106   I 

o5 

696153 

5 

28 

3o3847 

35 

26 

6485 T 2 

4 

23 

952043   I 

o5 

696470 

5 

28 

3o353o 

34 

3 

648766 

4- 

23 

9519S0   1 

o5 

696787 

5 

28 

3o32!3 

33 

649020 

4 

23 

951917   I 

o5 

697103 

5 

28 

302897 

3023Ho 

32 

^9 

649274 

4 

22 

95i§54   I 

o5 

697420 

5 

27 

3i 

3o 

649527 

4 

22 

951791    I 

o5 

697736 

5 

27 

302264 

3d 

3i 

9-649781 

4 

22 

9-951728   I 

o5 

9-698053 

5 

27 

10.301947 

29 

32 

65oo34 

4 

22 

95i665   I 

o5 

698169 
69S685 

5 

27 

3oi63i 

28 

33 

650287 

4 

21 

951602   1 

o5 

5 

26 

3oi3i5 

27 

34 

65o539 

4 

21 

95i539   I 

o5 

699001 

5 

26 

300999 

26 

35 

650792 

4 

21 

951476   I 

o5 

6993 1 6 

5 

26 

3oot)»4 

25 

36 

651044 

4 

20 

951412   I 

o5 

699632 

5 

26 

3oo368 

24 

3? 
38 

601297 

4 

20 

951349   « 

06 

699947 

5 

26 

3ooo53 

23 

65 1 549 

4 

20 

951286   I 

06 

700263 

5 

25 

299737 

22 

39 

65 1800 

4 

•9 

951222   I 

06 

700578 

5 

25 

299422 

21 

40 

652o52 

4 

•9 

951159   I 

06 

7CJ893 

5 

25 

299107 

20 

4i 

9-6523o4 

4 

ig 

9-951096   I 

06 

9-701208 

5 

24 

!0^ 298792 

19 

42 

652555 

4 

18 

95io32   1 

06 

701523 

5 

24 

298477 

18 

43 

652806 

4 

18 

930068   I 

06 

701837 

5 

24 

J98163 

17 

44 

653o57 

4 

18 

930905   I 
950841   1 

06 

702152 

5 

24 

297848 

16 

45 

6533o8 

4 

18 

06 

702466 

5 

24 

297534 

i5 

46 

653558 

4 

17 

950778   I 

06 

702780 

5 

23 

297220 

14 

47 

6538o8 

4 

17 

950714   I 

06 

703095 

5 

23 

206905 

i3 

48 

654059 

4 

17 

95o65o   I 

06 

703409 

5 

23 

290391 

13 

49 

654309 

4 

16 

95o586   I 

06 

703723 

5 

23 

296277 

II 

5o 

654558 

4 

16 

95o522   I 

07 

7o4o36 

5 

22 

293964 

10 

5i 

9.654808 

4 

16 

9-950458   I 

07 

9-704350 

5 

•22 

10^295650 

0 

53 

655o58 

4 

16 

95o394   I 

07 

704663 

5 

-22 

295337 

8 

53 

655307 

4 

i5 

95o33o   I 

07 

704977 

5 

•22 

295023 

7 

54 

655556 

4 

i5 

950266   I 

.07 

705290 

5 

-22 

294710 

5 

55 

6558o5 

4 

i5 

950202   I 

07 

7o56o3 

5 

•21 

294397 
294084 

5 

56 

656o54 

4 

14 

9501 38   I 

■  07 

705916 

5 

■21 

4 

S 

656302 

4 

•  14 

950074   I 

.07 

706 J 28 

5 

•21 

293772 

3 

656551 

4 

-14 

950010   I 

-07 

706541 

5 

-21 

93459 

3 

59 

656799 

4 

-i3 

949945   I 

-07 

706354 

5 

•21 

293 1 46 

I 

6o 



657047 

4-i3 

949881   I 

-07 

707166 

5-20 

292834 

t 

Cosuie 

D. 

Sine     I 

). 

Cotang. 

D. 

Tan.ar. 

M. 

(C3  DSOREE3.) 


SINES  AND  TANGENTS.      (27    DEGREES.) 


45 


a. 

0 

Slue 

D. 

Codino 

D 

Taug. 

D. 

Cotmig. 

60 

9.657047 

4-i3 

9-949881 

1-07 

9.707166 

5.20 

10.292834 

I 

657293 

4-i3 

949816 

1-07 

707478 

5-20 

292522   3^ 

2 

637342 

4-12 

949752 

1-07 

707790 

5-20 

292210 

33 

3 

637790 

4-12 

949688 

1-08 

708102 

5-20 

291898 

^2 

4 

658o37 

4-12 

949623 

i-oS 

708414 

5-19 

29 1 586 

56 

5 

658284 

4-12 

949558 

i-o8 

708726 

5-19 

291274 

55 

6 

658531 

4ii 

949494 

I -08 

709037 

5-19 

290963   L>4 

I 

658778 

4-u 

949429 

i-o3 

709349 

5-19 

290651   53 

639023 

4-II 

949364 

i-o8 

709660 

5-19 

290340 

32 

9 

659271 

4-10 

949300 

i-o3 

709971 

5-18 

290029 

5i 

IC 

63951- 

4-10 

949235 

i-o8 

710282 

5-18 

j 

289718 

5o 

11 

9-659763 

4-10 

9-949170 

1-08 

9-710593 

5-18 

10-289407 

S 

13 

660009 

4-09 

949105 

i-o3 

710904 

5-18 

289096 

i3 

^60255 

4-09 

949040 

1-08 

■71 121 5 

5-18 

288785 

47 

14 

660301 

4-09 

948975 

I -08 

7ii525 

5-17 

288475 

46 

i5 

660746 

4-09 

948910 

i-c8 

711836 

^'7 

288164 

45 

i6 

660991 

4-o8 

948845 

I -08 

712146 

5-17 

287854 

44 

17 

6612J6 

4-o8 

948780 

1-09 

712436 

5-17 

287544 

43 

i8 

661481 

4-o8 

948715 

I  -09 

712766 

5-16 

287234 

42 

'9 

661726 

4-07 

948650 

1  -09 

713076 

5-16 

286924 

41 

20 

661970 

4-07 

948584 

1-09 

713386 

5.16 

286614 

40 

21 

9-662214 

4-07 

9-948519 

1-09 

9-713696 

5.16 

io.2863o4 

39 

22 

662459 

4-07 

948454 

1-09 

714005 

5.16 

285995 

38 

23 

662 70 J 

4-o6 

948388 

1-09 

714314 

5.i5 

285686 

37 

24 

662946 

4-o6 

948323 

1-09 

714624 

5.i5 

285376 

36 

23 

663100 
66J4J3 

4-o6 

948257 

1-09 

714933 

5.i5 

283067 

35 

26 

4-o5 

948192 

1-09 

713242 

5-i5 

284758 

34 

27 

663677 

4-o5 

948126 

1-09 

7i555i 

5-14 

284449 

33 

28 

66J920 

4-o5 

948060 

1-09 

715860 

5-14 

284140 

32 

29 

664163 

4-o5 

947995 

I-IO 

716168 

5-14 

283832 

3i 

So 

664406 

4-04 

947929 

I-IO 

7 '6477 

5.14 

283323 

3o 

3i 

9-664648 

4-04 

J. 947863 

I-IO 

9-716785 

5.14 

10-283215 

'2 

32 

664891 

4-04 

■ '  947797 

1-IC 

717093 

5.i3 

282907 

28 

33 

665 1 33 

4-»3 

947731 

I-IO 

717401 

5-i3 

282399 

27 

34 

663375 

4-o3 

947665 

l-IO 

717709 

5-i3 

282291 

26 

35 

665617 

4-o3 

947600 

I-.O 

718017 

5-i3 

281983 

25 

36 

663839 

4-02 

947533 

l-IO 

718325 

5.i3 

281670 

24 

ll 

666100 

4-02 

947467 

110 

718633 

5-12 

281367 

23 

666342 

4-02 

947401 

l-IO 

7 1 8940 

5-12 

281060 

22 

39 

666583 

4-02 

947335 

I-IO 

719248 

5-12 

280732 

21 

40 

666824 

4-01 

947269 

I-IO 

719555 

5.12 

280445 

20 

41 

9-667065 

4-01 

9-947203 

I-IO 

9-719862 

5.12 

io.28oi38 

19 

42 

667305 

4-01 

947136 

I-U 

720169 

5-11 

27983. 

18 

43 

667346 

4-ci 

947070 

l-II 

720476 

5-II 

279324 

17 

44 

667786 

4-o{ 

947004 

l-II 

720783 

5. II 

279217 

16 

43 

6O0027 

4-00 

946937 

l-II 

721089 

5-11 

27:^911 

i5 

46 

66S267 

4-00 

946871 

I-II 

721396 

5.11 

278604 

14 

4T 

068306 

3-99 

946S04 

I-II 

721702 

5-10 

278298 

i3 

48 

668746 

^■99 

946738 

l-II 

722005 

5  10 

277991 

12 

49 

668986 

,"99 

946671 

I-II 

7223l3 

5-10 

277685 

II 

5o 

669223 

3-99 

946604 

I-II 

722621 

5.10 

277379 

10 

5i 

9-66.;i64 

3-g3 

g-946538 

I-II 

9.722927 

5-10 

10-277073 

I 

5s 

669703 

3-98 

94647' 

l-II 

723232 

5-09 

276768 

53 

609942 

3-98 

946404 

I-II 

723538 

5-09 

276462 

1 

5.4 

670181 

3-97 

946337 

I-II 

723844 

5.09 

276156 

6 

55 

670419 

3-97 

946270 

I-I2 

7241/9 

5-09 

275851 

5 

56 

670658 

3  97 

946203 

1-12 

724454 

5.09 

275546 

4 

5^ 

670896 

3-97 

946136 

I-I2 

724759 

5 -08 

275241 

3 

671134 

3-96 

946069 

1-12 

725o63 

5-o8 

274935 

2 

59 

671372 

3-96 

946002 

1-12 

725369 

5-08 

274631 

I 

66 

671609 

3-96 

945935 

I-I2 

725674 

5-08 

274326 

0 

CoBia« 

D. 

Sine 

D. 

Cotang. 

D. 

^I'lmg- 

M. 

(62    DEGREES.) 


46 


(28    DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

Sino 

D. 

Cosuie 

D. 

Tang. 

D. 

'l  Cotang. 
10-274326 

60 

0 

9-671609 

3-96 

9-945985 

I-I2 

9-725674 

5-08 

I 

671847 

3 

-95 

943868 

I-I2 

725979 

5 

-08 

274021 

59 

3 

672084 

3 

-95 

945800 

1-12 

726284 

5 

-07 

273716 

58 

3 

672321 

3 

.95 

943733 

I-I2 

726588 

5 

•  07 

278412 

57 

4 

67:553 

3 

-95 

945666 

I  -12 

726892 

5 

-07 

273108 

56 

5 

672795 
673032 

3 

•94 

945598 

1-12 

727197 

5 

-07 

272S03 

i'j 

6 

3 

•94 

945581 

I  -12 

727301 

5 

-07 

272499 

54 

5 

678268 

3 

•94 

945464 

I-l8 

727805 

5 

-Ot) 

272195 

53 

6735o5 

3 

-94 

945896 

l-l3 

728109 

5 

-06 

271891 

52 

9 

673s74i 

3 

93 

943828 

I-18 

728412 

5 

■06 

271583 

31 

10 

673977 

3 

93 

943261 

I-l3 

728716 

5 

-06 

271284 

30 

II 

9.6742I3 

3 

98 

9-945198 

1-18 

9-729020 

e. 

-06 

JO -270^80 

'4Q 

12 

674448 

3 

92 

943125 

I-18 

729828 

5 

o5 

270677 

48 

i3 

67.1684 

3 

92 

945038 

I-18 

729626 

5 

03 

270874 

47 

U 

674919 

3 

92 

944990 

1-18 

720929 

5 

o5 

270071 

46 

i5 

675155 

3 

92 

944922 

1-18 

780288 

5 

o5 

269767 

45 

i6 

675390 

3 

9" 

944854 

1-18 

780535 

5 

o5 

269165 

44 

\l 

675624 

3 

9' 

944786 

.  1-18 

780888 

5 

04 

269162 

43 

675859 

3 

9' 

944718 

1-18 

781141 

5 

04 

26SS,iQ 

42 

'9 

676094 

3 

91 

944650 

1-18 

731444 

5 

04 

268556 

41 

JO 

676328 

3 

90 

944382 

i-U 

781746 

5 

04 

268234 

40 

21 

9-676562 

3 

90 

9-9U5i4 

I-I4 

;  782048 

5 

04 

10-267952 

39 

22 

676796 

8 

90 

944446 

1-14 

782851 

5 

o3 

267649 

38 

23 

677080 

8 

90 

944877 

I-I4 

732658 

5 

08 

267347 

37 

24 

677264 

3 

89 

944809 

1-14 

782955 

5 

08 

267043 

36 

25 

677408 
677781 

3 

89 

944241 

1-14 

788257 

5 

o3 

266743 

35 

26 

8 

89 

944172 

1-14 

788558 

5 

o3 

266442 

34 

27 

677964 

3 

88 

944104 

1-14 

788860 

5 

02 

266140 

33 

28 

678197 

8 

88 

944086 

1-14 

784162 

5 

02 

265833 

32 

29 

678480 

3 

88 

948967 

1-14 

784463 

5 

02 

263387 

3i 

3o 

678668 

3 

83 

948899 

1-14 

734764 

5 

02 

265236 

3o 

3i 

9-678895 

3 

S7 

9-948880 

1-14 

9-785o66 

5 

03 

10-264934 

29 

32 

679123 

3 

^7 

948761 

1-14 

783867 

5 

02 

264688 

28 

33 

679860 

8 

«7 

943698 

i-i5 

735668 

5 

01 

264882 

27 

34 

679302 

3 

H7 

948624 

1-13 

783969 

5 

01 

26408 1 

26 

35 

619824 

8 

86 

9 ',8555 

i-i5 

786269 

5 

01 

268731 

25 

36 

68oo56 

3 

86 

948486 

i-i5 

786370 

5 

01 

268  i8o 

24 

ll 

680288 

8 

86 

943417 

i-i5 

786871 

5 

01 

268129 

23 

68o5i9 

8 

85 

943348 

i-i5 

787171 

5 

00 

262829 

22 

39 

680750 

3 

85 

948279 

i-i5 

787471 

5 

00 

262329 

21 

40 

680982 

3 

85 

948^10 

i-i5 

78777' 

5 

00 

262229 

20 

,  41 

9-681218 

3 

85 

9-943141 

i-i5 

9-738071 

5 

00 

10-261929 

IQ 

42 

681448 

3- 

84 

948072 

i-i5 

788871 

5 

00 

261629 

18 

43 

681674 

3 

84 

9',8oo3 

I  •  i5 

788671 

4 

99 

261829 

'7 

44 

681905 

3 

84 

942984 

:-i5 

788971 

4 

99 

261029 

16 

45 

682135 

8 

84 

94i864 

I -13 

789271 

4- 

99 

260729 

i5 

46 

6S2865 

8 

83 

942795 

1-16 

789570 

4 

99 

260480 

14 

47 

682595 

3 

88 

942726 

1-16 

789870 

4 

99 

260 i3o 

i3 

48 

6828:5 

3- 

88 

942')36 

1-16 

740169 

4 

90 

239881 

2  1 

19 

683o55 

3- 

88 

942387 

1-16 

740468 

4 

98 

239332 

II  1 

5o 

688284  , 

3- 

82 

942517 

1-16 

740767 

4 

93 

259288 

10  .' 

5 

r,-6835u 

3- 

82 

9-942448 

i-i6 

9-741066 

4 

93 

13  258984 

c 

5a 

688743 

8- 

82 

942878 

1-16 

741863 

4 

98 

258635 

8 

53 

688972 

8- 

82 

942808 

1-16 

741664 

4 

98 

258886 

7 

54 

684201 

3- 

81 

942289 

1-16 

741962 

4 

97 

258o33 

6 

55 

684480 

3 

81 

942169 

1-16 

742261 

4 

97 

257789 

5 

56 

684658 

3 

81 

942099 

1-16 

742359 

4- 

97 

237441 

4 

ll 

684887 

3 

80 

942029 

1-16 

742858 

4- 

97 

257142 

3 

685ii5 

3 

8q 

941939 

I -16 

7481 56 

4- 

97 

256844 

2 

59 

685343 

3 

80 

941889 

1-17 

748434 

4- 

07 

256546 

I 

60 

685571 

3-8o  , 

941819 

1-17 

7i3752 

4-96 

236248 

0 

CoBina 

D. 

Sine 

D. 

Cotanff. 

E 

..   1 

Tan?. 

(Gl   DEGREKa.) 


SIXES   AND   TAXGENTS.      (29    DEGREES.; 


47 


( 


M. 

Sine 

D. 

Cosine 

D. 

Tung. 

D. 

Cotung. 

1 

0 

9-685571 

3-8o 

9-941819 

1-17 

9-743752 

4-96 

10-256248 

60 

I 

685799 

3- 

79 

941749 

1-17 

744o5o 

4 

96 

255930  '  So 

a 

686027 

3- 

79 

941679 

1-17 

744348 

4 

96 

255652  1  58 

3 

686254 

3- 

79 

9 'i  1 609 

1-17 

74 '1645 

4 

96 

2557  55  1  57 

4 

686482 

3- 

]l 

941539 

I  - 17 

744943 

4 

96 

255o57  i  56 

5 

686709 

3 

941460 

I  -17 

7452.10 

4 

96 

254760  55 

6 

6369J6 

3 

78 

941398 

1-17 

745538 

4 

93 

234462  5i 

I 

687163 

3 

78 

941328 

I  - 17 

745835 

4 

93 

234165   53 

687389 

3 

78 

941258 

1-17 

746132 

4 

95 

253868 

52 

9 

687616 

3- 

77 

941187 

1-17 

746429 

4 

93 

253571 

5i 

10 

687843 

3 

77 

941117 

1-17 

746726 

4 

93 

253274 

5o 

II 

9-6S8o69 

3 

77 

9-941046 

i-iS 

9-747023 

4 

94 

10-252977 

it 

12 

688290 

3 

77 

940975 

1-18 

7473:9 

4 

94 

252&8I 

i3 

688521 

3 

76 

94ono5 

1-18 

747616 

4 

94 

2523S4 

47 

14 

68S747 

3 

76 

940834 

1-18 

7479^3 

4 

94 

2520S7 

46 

i5 

68S972 

3 

76 

940763 

1-18 

748209 

4 

94 

251791 

45 

i6 

689198 

3 

76 

940693 

1-18 

7485o5 

4 

93 

231495 

44 

17 

6S9423 

3 

73 

940622 

i-i8 

748801 

4 

93 

25 11 99 

43 

i8 

6S9648 

3 

7-T 

g4o55i 

i-i8 

749097 

4 

o3 

230933 

42 

>9 

689873 

3 

75 

9404S0 

1-18 

749393 

4 

93 

25o6o7 

41 

20 

690098 

3 

75 

940409 

i-i8 

749689 

4 

93 

25o3ii 

40 

21 

9-690323 

3 

74 

9.940338 

1-18 

9-7499^5 

4 

93 

io-25ooi5 

39 

22 

690548 

3 

74 

94--!67 

I -18 

730281 

4 

92 

2497 '9 

38 

23 

690772 

3 

74 

940196 

1-18 

730576 

4 

92 

249424 

37 

24 

690996 

3 

74 

940125 

1-19 

750S72 

4 

92 

249128 

36 

25 

691220 

3 

73 

940054 

1-19 

731 167 

4 

92 

248833 

35 

26 

691444 

3 

73 

9399S2 

1-19 

751462 

4 

92 

248538 

34 

11 

69 1 668 

3 

73 

93991 1 

1-19 

751757 

4 

92 

248243 

33 

691 89: 

3 

73 

939840 

1-19 

732052 

4 

9' 

2-i79'^ 

32 

29 

6921 i5 

3 

72 

939168 

1-19 

732347 

4 

91 

247613 

3i 

So 

692339 

3 

72 

939697 

1-19 

752642 

4 

91 

247358 

3o 

3i 

9-692562 

3 

72 

0-939625 

1-19 

9-752037 

4 

91 

10-247063 

29 

32 

692785 

3 

71 

'939554 

1-19 

753231 

4 

91 

2ihfj() 

28 

33 

693008 

3 

71 

939482 

1-19 

753526 

4 

91 

246474 

27 

34 

693231 

3 

71 

939410 

I-I9 

753820 

4 

90 

246180 

26 

35 

69J453 

3 

71 

939339 

I-I9 

7541 1 5 

4 

90 

245885 

23 

36 

693676 

3 

70 

939267 

1-20 

754409 
754703 

4 

90 

245591 

24 

u 

693'S98 

3 

70 

939195 

1-20 

4 

90 

245297 

23 

694120 

3 

70 

939123 

1-20 

754997 

4 

90 

24  ')oo3 

22 

39 

694342 

3 

70 

939052 

1-20 

755291 

4 

90 

24  '.709 

21 

40 

694564 

3 

69 

93S980 

1-20 

755585 

4 

89 

244413 

20 

41 

9-694786 

3 

69 

9-938908 

1-20 

9.755S78 

4 

89 

10-244122 

19 

42 

695007 

3 

69 

938836 

1-20 

756172 

4 

89 

24382S 

i3 

43 

695229 

•> 

69 

Qiori:j 

1-20 

756465 

4 

^9 

243535 

«7 

44 

695450 

3 

68 

938691 

1-20 

736759 

4 

^ 

243241 

16 

45 

695671 

3 

63 

93S619 

I -20 

757052 

4 

^9 

242918 

i5 

46 

693893 

3 

68 

938547 

1-20 

737345 

4 

88 

242655 

14 

^l 

696113 

3 

68 

938475 

1-20 

757638 

4 

88 

242362 

i3 

4S 

690334 

3 

67 

938402 

I-2I 

757931 

4 

88 

242069 

12 

49 

696554 

3 

67 

938330 

I-2I 

75.8224 

4 

88 

241776 

u 

be 

696775 

3 

67 

938258 

I-2I 

758317 

4.88 

2  11483 

10 

ifi 

9-696995 

3 

67 

9-938185 

I-2I 

9-738810 

4.88 

10-241190 

9 

55 

697215 

3 

66 

9381 i3 

I-2I 

759102 

4-87 

240898 

8 

53 

697435 

3 

66 

938040 

I  -21 

759395 

4.87 

240605 

7 

54 

697654 

3 

66 

937967 

I-2I 

739687 

4-87 

24o3i3 

6 

55 

^?7^74 

3 

-66 

937895 

I-2I 

759979 

4-87 

240021 

5 

56 

69.'509 1 

3 

.65 

937822 

I-2I 

760272 

4-87 

239728 

4 

^2 

698313 

3 

-65 

937749 

I-2I 

760364 

4-87 

239436 

3 

58 

698532 

3 

-65 

937676 

I-2I 

760856 

4-86 

239144 

2 

^ 

698751 

3 

-65 

937604 

I -2! 

761 148 

4-86 

238852 

I 

60 

698970 

3.64 

937531 

I-2I 

761439 

4-86 

238561 

0 

Cosine 

D. 

Sine 

D. 

Cotanor. 

D. 

Tao^- 

M. 

28 


(60    DEORKEfl.) 


I  i 


i8 

(3C 

DEGREES.;   A  TABLE 

OF  LOGARirnMIO 

0 

81110 

D. 

Cosine  |  I). 

1 

Tang. 

B. 

Cotang. 

q -698970 

3-64 

9-937531  j  I 

•21 

9-761439 

4-86 

io^23856i   5o 

I 

699189 

3 

-64 

937458   1 

•22 

761731 

4^86 

238269   59 
237977   58 

a 

699407 

3 

-64 

937335   1 

-22 

762023 

4^86 

3 

699626 

3 

-64 

937312   I 

-22 

762314 

4^86 

23  ;686  57 

.  A 

699844 

3 

-63 

937238   I 

•22 

762606 

4^85 

237394  !  56 

5 

700062 

3 

-63 

937165   I 

-22 

762807 

4-85 

237103   55 

6 

700280 

3 

-63 

937092   1 

-22 

763188 

4-85 

236SI2  '  54 

7 

700498 

3 

-63 

937019   I 

-22 

7634T9 

4-85 

236521  i  53  ' 

8 

7007 1 6 

3 

63 

936046   I 

-22 

763770 

4-85 

236230 

52 

9 

700933 

3 

62 

936872   I 

-22 

764061 

4-85 

230930 

5i 

10 

701101 

3 

62 

936799   1 

•22 

764352 

4-84 

23564^ 

5o 

II 

9-701368 

3 

62 

9-936725   I 

•22 

0  764643 

4-84 

io^235357 

49 

12 

7oi5S5 

3 

62 

936652   I 

•23 

764933 

4-84 

235067 

48 

i3 

701802 

3 

61 

936578   I 

•23 

765224 

4-84 

234776 

47 

U 

702019 

3 

61 

9305o5   I 

•23 

765514 

4-84 

234486 

46 

i5 

702235 

3 

61 

936431   I 

•23 

7658o5 

4-84 

234195 

45 

i6 

702402 

i 

ti 

936357   I 

•23 

766095 

4-84 

233905 

44 

17 

702669 
7028;i5 

3 

60 

Q36284   1 

•23 

766385 

4-83 

23361 5 

43 

i8 

3 

60 

936210   I 

23 

766675 

4-83 

233325 

42 

'9 

7o3ioi 

3 

60 

936i36   I 

23 

766965 

4^83 

233o35  41  1 

20 

703317 

3 

60 

936062   I 

23 

767255 

4-83 

232745 

40 

21 

9-703533 

3 

59 

9-935988   I 

23 

9-767545 

4-83 

10.232455 

39 

22 

703749 

3 

.59 

935914   I 

23 

767834 

4-83 

232166 

38 

23 

703964 

3 

59 

930840   1 

23 

768124 

4-82 

231876 

37 

24 

704179 

3 

59 

935766   I 

24 

76S413 

4-82 

23 1 587 

36 

25 

704395 

3 

59 

935692    ! 

24 

768703 

4-82 

231297 

35 

26 

704610 

3 

58 

935618    I 

24 

768992 

4-82 

23 1008 

34 

27 

704825 

3 

58 

935543    I 

24 

7692S1 

4-82 

230719 

33 

28 

700040 

3 

58 

935469    I 

24 

769070 

4-82 

23o43o 

32 

29 

705254 

3 

58 

955395    I 

24 

769860 

4-81 

23oi4o 

3i 

3o 

705469 

3 

57 

935320    I 

24 

770148 

4-8i 

22q852 

3o 

3i 

9-705683 

3 

^7 

9-935246    I 

24 

9-770437 

4-8i 

10-229563 

29 

32 

700898 

3 

57 

935171     1 

24 

770726 

4-8i 

229274 

28 

33 

7061 12 

3 

57 

935097    I 

24 

771015 

4-8i 

22S985 

27 

34 

706326 

3 

56 

935022     I 

24 

77i3o3 

4-8i 

228697 

26 

35 

706539 

3- 

56 

934948    I 

24 

771592 

4-81 

228408 

25 

36 

706753 

3- 

56 

934873    I 

24 

771880 

4-8o 

228120 

24 

37 

38 

706967 

3- 

56 

934798    ' 

20 

772168 

4-80 

227832 

23 

707180 

3- 

55 

934723    I 

25 

772457 

4-80 

227543 

22 

39 

707393 

3- 

55 

934649    I 

25 

772745 

4-80 

227255 

21 

40 

707606 

3- 

55 

934574    I 

25 

773o33 

4-80 

2264*7 

20 

41 

9-707819 

3- 

55 

9-934499    ' 

25 

9-773321 

4-80 

10. J  26679 

10 

^'i 

708032 

3- 

54 

934424    I 

25 

773608 

4-79 

226392 

18 

43 

708245 

3- 

54 

934349    I 

25 

773806 
774184 

4-79 

226104 

[I 

44 

708408 

3- 

54 

934274    I 

25 

4-79 

2a58i6 

45 

708670 

3- 

54 

934199    I 

25 

774471 

4-79 

225529 

i5 

46 

708882 

3- 

53 

934123    I- 

25 

774759 

4-79 

J2524I 

14 

8 

709094 

3- 

53 

934048    I 

25 

775o4f> 

4-79 

224954 

i3 

709306 

3- 

53 

933973    1 

25 

775333 

4-79 

224667 

la 

49 

709018 

3- 

53 

933898    1 

26 

770621 

4-78 

224379 

n 

5o 

709730 

3- 

53 

933822    1 

26 

775908 

4-78 

224092 

10 

5i 

9-7oo-)4i 

3- 

02 

9-933747    I- 

26 

9-776195 

4^78 

io.2238o5 

9 

52 

7 101 53 

3. 

52 

933671     I- 

26 

776482 

^•78 

2230l8 

8 

53 

710364 

3- 

52 

933596    I  • 

26 

776769 

4-78 

22333l 

7 

54 

710075 

3- 

52 

933520    I- 

26 

777050 

4-78 

222940  , 

6 

55 

710786 

3- 

5i 

933445    I- 

26 

777342 

4-78 

222608 

5 

56 

710997 

3- 

5i 

933360    I  - 
933293    1- 

26 

777628 

4-77 

222372 

4 

u 

71 1208 

3- 

5i 

26 

7779'5 

4-77 

222080    3  1 

711419 

3- 

5i 

933217    I- 

26 

778201 

4-77 

221799 

22l5l2 

2 

^ 

71 1629 

3- 

5o 

933141    I- 

26 

778487 

4-77 

I 

60 

71 1839 

3-5o 

933066    I  - 

26 

778774 

4-77 

221226 

0 

M. 

Co6ino 

D. 

Sine     I] 

.  1 

CotaiiEr. 

D. 

Tunor. 

(no  Dr 

GRl 

5;r;s') 

SINES   AND  TANGENTS.      (31    DEGREES.) 


49 


M. 

'  0 

Siue 

1).   j 

Cosine 

D.   j 

Tan?.   1 

D. 

Cotan». 

■  — 1 
60 

9-7ii83g 

3-5o 

9-933066 

1-26 

9-778774 

4-77 

10-221226 

I 

7i2o5o 

3-5o 

932990 

1-27 

779060 

4- 

77 

220940   39  1 

3 

712260 

3.5o 

932914 
932838 

1-27 

779346 

4- 

76 

220634 

58  1 

3 

712469 

3-40  1 

1-27 

779632 

4- 

76 

220368 

57 

4 

712679 

3-49  1 

932762 

'•27 

7799'8 

4- 

76 

220082 

56  1 

5 

7128S9 

3-49  1 

932685 

1-27 

780203 

4- 

76 

219797 

55 

0 

713098 

3-49  1 

932609 

1-27 

780489 

4- 

76 

2193:1 

54 

I  "1  i 

7i33u8 

3-49  ' 

932333 

1-27 

780775 

4- 

76 

219223 

53 

S  i 

»i35i7 

3-4H 

032437 

••27 

781060 

4- 

76 

218940 

52 

9 

71372c 

3-48 

932380 

1-27 

781346 

4- 

75 

218634 

57 

10 

713935 

3-48 

932304 

1-27 

781631 

4- 

75 

218369 

5o 

1 

II 

9-714144 

3-4S  j 

9-932228 

1-27 

9-781916 

4- 

75 

10-218084 

49 

12 

714352 

3-47  ' 

93h5i 

1-27 

782201 

4- 

75 

217799 

43 

i3 

714361 

3-47  1 

932073 

1-28 

782486 

4- 

75 

217514 

47 

U 

714769 

3-47  ' 

931998 

1-28 

782771 

4- 

75 

217229 

46 

i5 

7'497« 

3-47 

931921 

1-28 

783o56 

4- 

75 

216944 

45 

i6 

713186 

3-47 

931845 

1-28 

783341 

4- 

75 

216659 

44 

\l 

715394 

3-46 

931768 

1-28 

783626 

4- 

74 

216374 

43 

7i56o2 

3-46 

931691 

1-28 

783910 

4- 

74 

2 1 6090 

42 

>9 

7 1 5809 

3-46 

931614 

1-28 

784195 

4- 

74 

2i58o5 

41 

20 

716017 

3-46 

931 537 

1.28 

784479 

4- 

74 

215521 

40 

21 

9-716224 

3-45 

9-931460 

1.28 

9-784764 

4- 

74 

io-2i5236 

^2 

22 

716432 

3-45 

93 1 383 

1-28 

785048 

4- 

74 

214932 

3S 

23 

7 1 6639 

3-45 

93i3o6 

1.28 

783332 

4- 

73 

214668 

37 

24 

716846 

3-45 

931229 

1  -29 

7856i6 

4- 

73 

214384 

36 

25 

717053 

3-45 

93 1 1 52 

1-29 

785900 

4- 

73 

214100 

35 

26 

r:7259 

3-44 

931075 

1-29 

7S6184 

4- 

73 

2i38i6 

34 

^1 

717466 

3-44 

930998 

1-29 

786468 

4- 

73 

2:3532 

33 

28 

717673 

3-44 

93092 1 

1-29 

786752 

4- 

73 

213248 

32 

?9 

717879 

3-44 

930843 

1-29 

787036 

4- 

73 

212964 

3i 

3o 

718080 

3-43 

930766 

1-29 

787319 

4- 

72 

212681 

3o 

3i 

9-718291 

3-43 

9-930688 

I  -29 

9-787603 

4 

72 

10-212397 

20 

28 

32 

718497 

3-43 

-93061 1 

I  -29 

787886 

4 

72 

212114 

33 

718703 

3-43 

930333 

1-29 

788170 

4 

72 

2ii83o 

27 

34 

7 1 8909 

3-43 

930436 

1-29 

788453 

4 

72 

21 1 547 

26 

35 

719114 

3-42 

93o373 

1-29 

788736 

4 

72 

21 1264 

25 

36 

719320 

3-42 

93o3oo 

i-3o 

789019 

4 

72 

210981 

24 

37 

719325 

3-42 

93o223 

i-3o 

789302 

4 

71 

210698 

23 

38 

719730 

3-42 

930145 

i-3o 

789385 

4 

71 

2104:5 

22 

39 

719935 

3-41 

930067 

i-3o 

789868 

4 

71 

2IOl32 

21 

4o 

720140 

3-41 

929989 

i-3o 

790i5i 

4 

7« 

209849 

20 

4i 

9-720345 

3-41 

9-92Q9I1 

i.3o 

9-790433 

4 

71 

10-209567 

\t 

42 

720549 

3-41 

929833 

i-3o 

790716 

4 

71 

209284 

43 

720754 

3-40 

929755 

I  -30 

790999 

4 

7« 

209001 

n 

44 

720958 

3-40 

929677 

1-30 

791281 

4 

7« 

208719 

16 

45 

721162 

3-40 

929599 

i.3o 

791563 

4 

70 

208437 

i5 

46 

72 1 366 

3-40 

929321 

i-3o 

791846 

4 

70 

2081 54 

14 

47 

721370 

3-40 

929442 

i-3o 

792128 

4 

70 

207872 

i3 

48 

721774 

3-39 

929564 

1-31 

792410 

4 

70 

207590 

12 

f^ 

721978 

3-39 

929286 

i-3i 

792692 

4 

70 

207308 

II 

5o 

722181 

3-39 

929207 

i-3i 

792974 

4 

70 

207026 

10 

5i 

9-722385 

3-39 

9-929129 

i-3i 

9-793256 

4 

70 

10  206744 

9 

52 

722588 

3-39 

929030 
928072 
9:8893 

i-3i 

793538 

4 

69 

206462 

8 

53 

722791 

3-33 

i-3i 

793819 

4 

69 

206181 

7 

54 

722994 

3-38 

i-3i 

79i'0' 

4 

.69 

205899 

6 

55 

723197 

3-38 

928815 

i.3i 

794383 

4 

-69 

2o56i7 

3 

56 

723400 

3-38 

928736 

i-3i 

794664 

4 

.69 

203336 

4 

l^ 

7236o3 

3-37 

928637 

i-3i 

79 '.9  ^5 

4 

-69 

2o5o55 

3 

58 

7238o5 

3-37 

9283-8 

i-3i 

793227 

4 

.60 

204773 

a 

59 

724007 

3-37 

928499 

i-3i 

793508 

4 

-68 

204492 

I 

6o 

724210 

3-37 

928420 

i-3i 

793789 

4-68 

204211 

0 

1  Cosine 

D. 

1   Sine 

D. 

1  Cotane. 

D. 

Tanjj... 

U. 

(58    DEGREES.) 


50 


(32   DEGREES.)      A  TABLE   OF   LOGARITHMIC 


M.  I 

Tl 
2 1 

5  ! 

6  I 

7 
8 

'J 

10 

II 

12 

i3 
14 
i5 
i6 

17 
i8 

19 

20 

21 

22 
23 
24 
25 
26 

27 
28 

29 

3o 
3i 

32 

33 

34 
3n 
36 
37 
38 
39 
40 

41 
42 
43 
44 

45 
46 

47 

48 

^5^ 

5i 
5i 
53 
54 
55 
56 

57 
58 
59 
60 


Sine 

724210 

724412 
72461 i 
724H16 

7 230! 7 
J252'9 
725i20 

72 J622 
72  3823 
726024 
726225 

r-72642(- 

736626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 

I  728427 
72S626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 

i-73o4i5 
7306 1 3 
730811 
731009 
731206 
731404 
731602 
731799 
731996 
732193 

9-732390 
732587 
732784 
732980 

733177 
733373 
733569 
733763 
733961 
734157 

9.734353 

734549 
734744 

734939 
735 1 35 
735jJo 
735523 
735719 
733914 
736109 


Coshie 


Taui; 


Cosiro 


3-37 
3-37 
3-36 
3-36 
3-36 
3-36 
3-35 
3-35 
3-35 
3-35 
3-35 

3-34 
3-34 
3-34 
3.34 
3- -4 
3-33 
3-33 
3.33 
3-33 
3.33 

3-32 
3-32 
3-32 
3-32 
3-31 
3.31 
3-31 
3-31 
3-30 
3.30 

3-30 
3-30 
3.30 
3.29 
3.29 
3.29 
3-29 
3.29 
3.28 
3-28 

3.28 
3.28 
3-28 
3.27 
3-27 
3.27 
3.27 
3.27 
3-26 
3.26 

3-26 
3-26 
3-25 
3-25 
3-25 
3:5 
3-25 
3.24 
3-24 
3.24 


9  928420  I 
928342  I 
928263 
928183 
928104  • 
928025  I 
921946  j 
927867  I 

9277^^7 
927708 
927629 

9.927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
92691 1 
926831 

9  926751 
926671 
026591 
9265i I 
926431 
926351 
926270 
926190 
9261 10 
926029 

9  925949 
925868 
923788 
925707 
925626 
925545 
925465 
925384 
9253o3 
925222 

9-925I4I 
925o6o 

924979 
924897 
924816 
924735 
924654 
924572 

924491 
924409 

9.924328 
92424(3 
924164 
924083 
924001 
023919 
923837 
923755 
9236i3 
923591 


D. 


Sine 


1.32 

1-32 

1.32 

1-32 

1.32 

1-32 
1.32 

1.32 

1-32 

1.32 

1 .32 

1-32 

1-33 
1-33 
1.33 
1-33 
1.33 
1.33 
1.33 
1.33 
1.33 

1.33 
1.33 
1.33 
1-34 
1-34 
1.34 
1-34 
1-34 
1-34 
1-34 

1-^4 
1-4 
1-34 
1-34 
1.34 
1-35 
1 .35 
1.35 
1.35 
1.35 

1.35 
1.35 
1.35 
I  -35 
1.35 
1.36 
1.36 
1-36 
1.36 
1-36 

1.36 
1-36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.37 
1.37 
1.37 


L\ 


9.795789 
796070 
796351 
796632 
796913 
797194 
797473 
797733 
798036 
798316 
798596 

9.798817 
799157 
799437 
799717 
799997 
800277 
800557 
8oo836 
801116 
801396 

9.801675 
801955 
802234 

8n75l3 

802792 
808072 
8o335i 
8o363o 
803908 
804187 

9 . 804466 
8oi745 
8o5o23 
eo53o2 
8o558o 
8o5859 
806137 
80641 5 
806693 
806971 

9-807249 
807527 
807805 
808083 
8o836i 
8o8638 
80S916 
809 193 
80947 1 
809748 

9.810025 
8io3o2 
8io58o 
810857 
8iii34 
811410 
811607 
811964 
812241 
812517 


T). 


Cotansr. 


4.68 

4.68 
4-63 
4.68 
4-68 
4-68 
4.68 
4.68 
4-67 
4-67 
4-67 

4-67 
4-67 
4-67 
4-67 
4-66 
4-66 
4-66 
4-66 
4-66 
4-66 

4-66 
4-66 
4-65 
4-65 
4-65 
4-65 
4-65 
4-65 
4-65 
4-65 

4-64 
4-64 
4-64 
4-64 
4-64 
4-64 
4-64 
4-63 
4-63 
4-63 

4-63 
4-63 
4-63 

4-63 
4.62 
4-62 
4.62 
4-62 
4-62 

4-62 
4.62 
4.62 
4  62 
4.61 
4.61 
4.61 
4.61 
4.61 
4-61 


Cotaiig. 


60 


10-204211 
i?3o3o 

2o3ojiQ 
203368 
2030S7 
202806 

202525 

202243 
201964 
201684  i  5i 

201404  I  30 


54 

53 


10.201123 

200843 
2oo563 
200283 
2oooo3 
199723 
199443 
1 99 1 64 
198884 
198604 

10. 198325 

198045 
197766 
197487 
197208 
196928 
196649 
196370 
1 96092 
195813 

10.195534 
195255 

194977 
194698 
194420 
194141 
I93S63 
193585 
193307 
193029 

10-192751 
192473 
19:195 
19:917 
IQ1639 
19 1 362 
1 91 084 
190807 
190529 
19023J 

10-189975 
189698 
189420 
189143 
1 88866 
188390 
i883i3 
i8So36 
187759 
187483 


D. 


49 

48 

47 

46  ' 
45 

44 
43 
42 
41 
40 

39 

38 

37 
36 
35 
34 
33 

32 

3i 

3o 

It 

27 
26 

25 

24 

23 
22 
21 
20 


17 
16 

i5 
14 
i3 
:3 
II 
10 


Tang. 


M. 


(57  DEGREES.) 


SINES   AXL    TANGENTS       (33    DEGREES.) 


61 


"m. 

1   0 

Sine 

D. 

Cosine     D. 

Tang. 

D. 

Cotang. 

60 

9-736ioq 

3-24 

9-923391    I 

37 

9-8i25i7 

4-61 

10  187482 

I 

7363o3 

3 

24 

923309   I 

37 

8 1 279  J 

4 

61 

187206 

u 

3 

73^498 

3 

24 

923427   I 

37 

813070 

4 

61 

186930 

3 

730692 

3 

23 

923345   I 

37 

813347 

4 

60 

186653 

57 

4 

736886 

3 

23 

923263   I 

37 

8i3623 

4 

60 

180377 

56 

5 

737080 

3 

23 

923181   1 

37 

813899 

4 

60 

186101 

55 

5 

737:^4 

3 

23 

923098   I 

37 

814173 

4 

60 

185825 

54 

7 

7^7-^67 

3 

24 

923016   I 

37 

814432 

4 

60 

183548 

53 

8 

737661 

3 

22 

922933   1 

37 

814728 

4 

60 

185272 

52 

P 

737S55 

3 

22 

922831    I 

37 

8i5oo4 

4 

60 

184996 

5i 

10 

73S048 

3 

22 

922768   I 

38 

815279 

4 

60 

184721 

5o 

11 

9-738241 

3 

22 

9-922686   > 

38 

9-8i5555 

4 

59 

10-184445 

49 

(2 

738434 

3 

22 

9225o3    I 

38 

8i583i 

4 

5o 

1 84 1 69 

48 

i3 

738627 

8 

21 

922320    I 

38 

S16107 

4 

59 

183S93 

47 

14 

73.S820 

3 

21 

922438    I 

38 

815382 

4 

59 

1 836 18 

46 

i5 

739013 

3 

21 

922355    I 

38 

8i6653 

4 

59 

183342 

45 

i6 

739206 

3 

21 

922272    I 

38 

816933 

4 

59 

i83o07 

44 

17 

739398 

3 

21 

922189    I 

38 

817209 

4 

59 

182791 

43 

i8 

739090 

3 

20 

922106    I 

38 

817484 

4 

59 

i825i6 

42 

•9 

739783 

3 

20 

922023    I 

38 

817739 

4 

5n 

182241 

41 

to 

739973 

3 

20 

921940    I 

38 

8i8o33 

4 

58 

181965 

40 

ti 

9.740167 

3 

20 

9-921837    I 

39 

g-8i83io 

4 

58 

10-181690 

39 

22 

740359 

3 

20 

921774    I 

39 

8i8585 

4 

53 

181415 

38 

tS 

740330 

3 

19 

921691     I 

39 

818860 

4 

58 

181 140 

37 

H 

740742 

3 

'9 

921007    I 

39 

819135 

4 

58 

i8o865 

36 

25 

740934 

3 

19 

921524    I 

39 

819410 

4 

58 

180590 

35 

26 

741123 

3 

19 

921441     I 

39 

819684 

4 

58 

i8o3i6 

■34 

27 

741 3 16 

3 

921337    1 

39 

819959 

4 

58 

180041 

33 

28 

741 303 

3 

18 

921274    1 

39 

820234 

4 

58 

179766 

32 

S9 

74iC>99 

3 

18 

921 190    I 

39 

820308 

4 

57 

179492 

3i 

3o 

741889 

3 

18 

921107    I 

39 

820783 

4 

57 

179217 

3o 

3i 

9-742080 

3 

18 

9-921023    I 

39 

9-821057 

4 

57 

10-178943 

29 

32 

742271 

3 

18 

920939    I 

40 

821332 

4 

57 

178668 

28 

33 

742462 

3 

17 

920836    I 

40 

821606 

4 

57 

178394 

27 

34 

742632 

3 

17 

920772    I 

40 

821880 

4 

57 

178120 

26 

35 

742S42 

3 

17 

920688    I 

40 

822154 

4 

57 

177^46 

25 

35 

743o33 

3 

17 

920604    I 

40 

822429 
822703 

4 

57 

177571 

24 

37 

7432  23 

3 

'7 

920320    I 

40 

4 

57 

177297 

23 

38 

7434 1 3 

3 

16 

920436    I 

40 

822977 

4 

56 

177023 

22 

39 

743602 

3 

16 

920332    I 

40 

823230 

4 

56 

176750 

21 

40 

743792 

3 

16 

920268    I 

40 

823524 

4 

56 

176476 

20 

4t 

9-743982 

3 

16 

9-920184    I 

40 

9-823798 

4 

56 

10-176202 

'2 

42 

744171 

3 

16 

920099    1 
920013     I 

40 

824072 

4 

56 

175928 

18 

43 

744361 

3 

i5 

40 

824345 

4 

56 

173635 

17 

44 

744330 

3 

i5 

919031   I 
919846   I 

41 

824619 

4 

56 

173381 

16 

45 

744739 

3 

i5 

41 

8248v3 

4 

56 

173107 

i5 

46 

7449  ^f^ 

3 

i5 

919762   1 

41 

823166 

4 

56 

174334 

14 

8 

743117 

3 

i5 

919677   I 

41 

825439 

4 

55 

1 7436 1 

i3 

745306 

3 

14 

919393   I 

41 

825713 

4 

55 

174287 

12 

49 

743494 

3 

14 

919308   I 

41 

823986 

4 

55 

174014 

II 

5o 

743083 

3 

14 

919424   I 

41 

826239 

4 

55 

173741 

10 

5i 

9  ;4387i 

3 

14 

9-919339   I 

41 

9-826532 

4 

55 

10-173468 

I 

52 

746039 

3 

14 

919234   I 

41 

826805 

4 

55 

173195 

53 

746248 

3 

i3 

919169   I 

41 

827078 

4 

55 

172922 

I 

54 

746436 

4 

i3 

919085   I 

41 

827331 

4 

5:, 

172649 

55 

746624 

3 

i3 

919000   I 

41 

827624 

4 

5? 

172376 

5 

56 

746812 

3 

i3 

918913   I 

42 

827897 

4 

54 

172103 

4 

57 

746999 

3 

i3 

9i883o   I 

42 

828170 

4 

54 

171830 

3 

58 

747187 

3 

12 

918745   I 

42 

828442 

4 

54 

171553 

a 

59 

747374 

3 

12 

918659   I 

•42 

828715 

4 

54 

17128.'^ 

I 

6o 

747362 

3-12 

9.8574   I 

•42 

828987 

4-54 

171013 
Tanpf. 

0 

at. 

Cosine 

_IX 

Sine     I 

). 

Cot&ng. 

1 

3. 

(56    DEGREK8.) 


&2 


(34  DEGREES.)   A  TABLE  OF  LOGARITHMIC 


M.  { 

I  1 
?  i 

^  I 

4  I 

5  I 

6  I 

«i 

9  ! 

10 


II 

12 

i3 
14 
i5 
i6 

\l 

19 
20 

21 

23 

23 
24 
25 
26 

27 


Sine 


D. 


3i 

32 

33 
34 
35 
36 

37 
38 

39 
40 

41 

42 
43 
44 
45 
46 

47 
48 

49 
5o 

5i 

52 

53 

54 
55 
56 

u 

60 


747562 

747749 
747936 
74«i23 
74B3IO 
748497 
748683 
748870 
749056 
749243 
749429 

•749615 
749801 

7  49987 
730172 
75o358 
75o543 
750729 
730914 
75ioQ9 
75 1 -.84 

•751469 
75.6.'-,4 
751889 
75202} 
75220S 
752892 
752376 
752760 
752944 
753128 

•753312 

753495 
753679 
75J862 
754046 
754229 
754412 
734595 
754778 
754960 

•755143 
755326 
755508 
755690 
755872 
756054 
756236 
736418 
736600 
736782 

•756963 
757144 
757826 
757307 
7576S8 
757869 
758o5o 
758230 
758411 
753591 


Cosine 


12 
12 

12 

1 

I 

1 

I 

I 

10 

10 

10 

10 
10 
09 
09 
09 
09 

o^ 

08 
08 

08 
08 
08 
07 
07 
07 
07 
07 
06 
06 

06 

06 
06 

o5 
o5 
o5 

03 

o5 
04 
04 

04 
04 
04 
04 
o3 
o3 
o3 
o3 
o3 
02 

02 
02 
02 
02 
01 
01 
01 
01 
01 
01 


D. 


Cosine 

D.   1 

9-9'8574 

1-42 

918489 

1-42 

918404 

1-42 

9i83i8 

1-42 

918233 

1-42 

018147 

1-42 

918062 

1-42 

917976 

1-43 

917891 

1.43 

917805 

1.43 

917719 

1-43 

9-917^34 

1.43 

9.7548 

I  43 

917462 

1.43 

917376 

1-48 

917290 

1.43 

917204 

1-43 

917118 

1-44 

917082 

1-44 

9 1 6946 

1-44 

916839 

1-44 

9-916773 

1-44 

916687 

1-44 

9 1 6600 

1-44 

9i63i4 

1-44 

916427 

1-44 

916841 

1-44 

916234 

■  •44 

916167 

1-45 

9 1 608 1 

1^45 

915994 

1-45 

9-915907 

1.45 

915820 

1^45 

915733 

1.45 

915646 

1.45 

915559 

1.45 

915472 

1-45 

915385 

1-45 

915297 

1-45 

915210 

1-45 

915128 

1-46 

9-9i5o85 

..46 

914948 
914860 

1-46 
■  •46 

914773 

1-46 

914685 

1-46 

914598 

1-46 

914510 

1-46 

914422 

1-46 

914334 

1-46 

914246 

1-47 

9.914158 

1-47 

914070 

1-47 

918982 

1-47 

918894 

t-47 

918806 

1-47 

913718 

»  •  -M 

Qi368o 

913541 

1-47 

918453 

1-47 

918365 

1-47 

Sine 

D.  1 

Tung. 


9- 


9- 


9- 


■828987 
829260 
829532 
829803 
880077 
880849 
880621 
880898 
83ii65 
881487 
881709 

■881981 
882253 
882525 
882796 
888068 
883839 
83861 1 
838882 
884154 
834425 

884696 
884967 
885238 
835309 
883780 
886o5i 
83632  2 
886398 
886864 
887184 

8374o5 
887675 
887946 
888216 
888487 
888757 
889027 
889297 
889568 
889888 

■840108 
840878 
840647 
840917 
841187 
841457 
841726 
84 1 996 
842266 
842535 

842805 
848074 
843343 
848612 
843882 
844i5i 
844420 
844689 
844958 
843227 


D. 


54 
54 
54 
54 
54 
53 
53 
53 
53 
58 
53 

53 
53 
53 
53 

52 

32 
52 
52 
52 
52 

52 
52 
52 
52 

5i 

5i 
5i 
5i 
5i 
5i 

5i 
5i 
5i 
5i 
5o 
5o 
5o 
5o 
5o 
5o 

5o 
5o 
5o 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 

ti 

48 
48 
48 
48 
48 


Cotang. 


10 


10 


10' 


Ou'nntr-  i D^     Tang.  i_M.J 


71013 

70740 
70468 
70195 
69928 
69651 
69379 
69107 
68835 
68563 
6829! 

68019 

67747 
67475 
6720.J 
66982 
6606 1 
66889 
66118 
63846 
65575 

658o4 
65o83 
64762 
64491 
64220 
68949 
68678 
63407 
63i36 
62866 

62595 
62825 
62034 
61784 
6i5i3 
61243 
60978 
60708 
60482 
60162 

59892 
39622 
59853 
59088 
588 1 8 
58343 
38274 
58oo4 
37784 
57465 

57195 
56926 
56657 
56388 
56ii8 
55849 
55580 
553 1 1 
55o42 
54773 


60 
So 

'  58 
5- 

i  5& 

!  55 

'  54 

i  53 

52 

5i 
5o 

49  I 

:  48 

:  47 

46 
45 
44 
43 

i  ^2 

41 

i  40 

I  38 

I  37 
36 
35 

i  ^^  , 
I  33 

32 

3i 
3o 

29 

28 

27 
26 

25 

24 

23 
22 
21 
20 

\t 

17 
16 

i5 

14 

1: 
II 
10 


(56  DEGREES.) 


SINES   AND  TANGENTS.      (35    DEGREESJ 


63 


M. 

Sine 

D. 

Cosine 

I). 

Tang. 

D. 

Cotang. 

0 

9-75859! 

3-01 

9-913365 

1-47 

9-845227 

4-48 

to- 154773 

60 

I 

758772 

3 

00 

913276 

1-47 

845496 

4 

48 

1 54304 

59 

2 

758932 

3 

00 

9131S7 

1-48 

845764 

4 

48 

154286 

58 

J 

759132 

3 

00 

913099 

1-48 

846088 

4 

48 

153967 
153698 
1 53430 

57 

4 

759312 

3 

00 

9i3oio 

1-48 

846802 

4 

48 

5b 

'j 

759492 

3 

00 

912922 

1-43 

846570 

4 

47 

55 

6 

759672 

2 

99 

912833 

1-48 

846889 

4 

47 

i58i6i 

54 

I 

759852 

2 

99 

912744 

1-48 

847107 

4 

47 

152893 

53 

760031 

2 

99 

912655 

1-48 

847376 

4 

47 

152624 

52 

9 

760211 

2 

99 

912566 

1-43 

847644 

4 

47 

152856 

5i 

lO 

760800 

2 

99 

9'2477 

1-48 

847918 

4 

47 

152087 

5o 

II 

9.760569 

2 

98 

9-912388 

1-48 

9-848181 

4- 

47 

io-i5i8i9 

49 

II 

760748 

2 

93 

912299 

1-49 

848449 

4- 

47 

i5i35i 

48 

i3 

760927 

2 

98 

912210 

1-49 

848717 

4 

47 

i5i283 

47 

14 

761 106 

2 

98 

912121 

1-49 

848986 

4 

47 

i5ioi4 

46 

i5 

761285 

2- 

98 

912081 

1-49 

849234 

4 

47 

130746 

45 

i6 

761464 

2- 

98 

9119^2 

1-49 

849322 

4 

47 

150478 

44 

17 

761642 

2 

97 

911853 

1-49 

849790 

4 

46 

l5o2IO 

43 

i8 

76 1 82 1 

2 

97 

911768 

1-49 

850038 

4 

46 

149942 

42 

'9 

761999 

2 

97 

911674 

1-49 

85o325 

4 

46 

149675 

41 

20 

762177 

2 

97 

9ii584 

1-49 

830598 

4 

46 

149407 

40 

21 

9-762356 

2- 

97 

9-911495 

1-49 

9-85o86i 

4 

46 

10-149139 

^2 

22 

762534 

2 

96 

911405 

1-49 

85II29 

4 

46 

I48S7I 

38 

23 

762712 

2- 

96 

9ii3i5 

i-5o 

851896 

4 

46 

148604 

37 

24 

762889 

0  , 

96 

911226 

i-5o 

85 1 664 

4 

46 

148386 

36 

25 

763067 

2 

96 

911186 

i-5o 

851981 

4 

46 

148069 

35 

26 

763245 

2- 

96 

911046 

i-5o 

852199 

4 

46 

147801 

34 

27 

763422 

^  ' 

96 

910936 

i-5o 

852466 

4 

46 

147^34 

38 

28 

763600 

2 

95 

9ioe«6 

i-5o 

852733 

4 

45 

14-267 

32 

29 

763777 

2 

95 

910775 

i-5o 

838001 

4 

45 

146999 
146732 

8i 

3o 

763954 

2 

95 

9106S6 

i-5o 

858263 

4 

45 

3o 

3i 

9-764131 

2 

95 

9-910596 

i-5o 

9-853535 

4 

45 

10-146465 

39 

32 

764308 

2 

95 

9io5o6 

i-5o 

858802 

4 

45 

146198 

28 

33 

7644S5 

2 

94 

910413 

i-5o 

854069 

4 

45 

i459Ji 

27 

34 

764662 

2 

94 

910825 

i-5i 

854836 

4 

45 

143664 

26 

35 

764838 

2 

94 

910285 

i-5i 

854603 

4 

45 

145397 

25 

36 

765oi5 

2 

94 

910144 

i-5i 

854870 

4 

45 

143180 

24 

37 

765igi 

2 

94 

910054 

i-5i 

855187 

4 

45 

144868 

23 

38 

765367 

2 

94 

909968 

i-5i 

855404 

4 

45 

144596 

22 

39 

765544 

2 

93 

909878 

i-5i 

855671 

4 

44 

141329 

21 

4o 

765720 

J 

93 

909782 

i-5i 

855988 

4 

44 

144062 

20 

4i 

9-765896 

2 

93 

9-909691 

i-5i 

9-856204 

4 

44 

10-143796 

19 

42 

766072 

2 

93 

909601 

i-5i 

856471 

4 

44 

143529 
148268 

18 

43 

766247 

2 

93 

909510 

i-5i 

856787 

4 

44 

'7 

44 

766423 

2 

93 

9094 1 Q 

i-5i 

857004 

4 

44 

142996 
142780 

16 

45 

766598 

2 

92 

909328 

1-52 

857270 

4 

44 

i5 

46 

766774 

2 

92 

909287 

1-52 

857587 

4 

44 

142463 

14 

47 

766949 

2 

92 

909 1 46 

1-52 

857808 

4 

44 

14^197 

i3 

«E 

767124 

2 

92 

909055 

1-52 

358069 

4 

44 

141981 

12 

4? 

767300 

2 

92 

90^^:964 

1-52 

858386 

4 

44 

141664 

11 

5c 

767473 

2 

91 

90S878 

1-52 

85S602 

4 

43 

141398 

10 

5j 

9  767649 

2 

91 

9-908781 

1-52 

9-858863 

4 

43 

io-i4ii32 

t 

52 

767824 

2 

9' 

908690 

1-52 

859184 

4 

43 

140866 

53 

767999 

2 

9' 

908599 

1-52 

839400 

4 

43 

140600 

7 

54 

768173 

2 

91 

908507 

1  52 

359666 

4 

43 

140884 

5 

55 

768348 

2 

90 

908416 

1-53 

359932 

4 

43 

140068 

5 

56 

768522 

2 

90 

90S824 

1-53 

860198 

4 

43 

189802 

4 

S 

768697 

2 

90 

908288 

1-53 

860464 

4 

-43 

189536 

3 

768871 

2 

-90 

908141 

1-53 

860780 

4 

-43 

189270 

3 

59 

769045 

2 

.90 

908049 

1-53 

860995 

4 

.43 

189005 

I 

60 

769219 

2-90 

907958 

1-53 

861261 

4- 43 

138739 

0 

Cosine 

1  D. 

Sine 

D. 

Cotang. 

D. 

Tang. 

M. 

(54    DEGREES.) 


54 


(36   DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M.  1   Siue 

D. 

Cosiuo    I 

). 

Tf.ng. 

D. 

Cotang. 

o 

9-769219 

2 

-90 

9-907958   I 

53 

9-3ti26i 

4-43 

10  138739 

60 

I 

769.393 

2 

89 

907866   I 

53 

36i527 

4 

43 

138473 

S 

2 

769366 

2 

89 

907774   1 

53 

861792 

4 

42 

138208 

3 

769740 

2 

89 

907682   I 

53 

862058 

4 

42 

137942 

57 

4 

769913 

2 

89 

907590   I 

53 

862323 

4 

42 

13-677 

56 

5 

7700S7 

2 

89 

907498   1 

53 

862589 

4 

42 

i374ii 

55 

6 

770260 

2 

88 

907406   I 

53 

862854 

4 

42 

137146 

54 

770433 

2 

88 

907314   1 

54 

863 1 19 

4 

42 

1 3688 1 

5S 

770606 

2 

83 

907222   I 

54 

863383 

4 

47 

1 366 1 5 

52 

9 

770779 

2 

88 

907129   I 

54 

863650 

4 

4- 

1 36350 

5i 

10 

770952 

2 

88 

907037   I 

54 

863915 

4 

4 

i36o85 

5o 

II 

9-77II25 

2 

88 

9-906045   I 
906852   1 

54 

9.864180 

4 

Aj 

10-135820 

49 

12 

77129S 

2 

87 

54 

864445 

4 

d2 

135555 

48 

i3 

771470 

2 

87 

906760   I 

54 

864710 

4 

42 

135290 

47 

14 

771643 

2 

87 

906667   I 

54 

864975 

4 

41 

i35o25 

46 

i5 

771815 

2 

87 

906575   I 

54 

865240 

4 

41 

134760 

45 

i6 

771987 

2 

87 

9064^2   I 

54 

8655o5 

4 

41 

134495 

44 

\l 

772109 

2 

87 

906389   I 

55 

863770 

4 

41 

i3423o 

43 

772331 

2 

86 

906296   I 

55 

866o35 

4 

41 

133965 

42 

19 

772303 

2 

86 

906204   I 

55 

866300 

4 

4. 

I337O0 

41 

20 

772675 

2 

86 

9061 1 1   I, 

55 

866564 

4 

^i 

133436 

40 

21 

9-772847 

2 

86 

9-906018   I 

55 

9-866829 

4 

4f 

j-i33i7i 

39 

22 

773018 

2 

86 

903025     I 

905832   I 

55 

867094 

4 

4» 

i32Qo6 

38 

23 

773190 

2 

86 

55 

867358 

4 

4i 

132642 

37 

24 

773361 

2 

85 

903739   I 

55 

867623 

4 

41 

132377 

36 

25 

773533 

2 

85 

905643   I 

55 

867887 

4 

41 

I32ii3 

35 

26 

773704 

2 

85 

905552   I 

55 

8681 52 

4 

40 

i3i«48 

34 

11 

773875 

2 

85 

905439   1 

55 

868416 

4 

40 

13 1 584 

33 

774046 

2 

85 

905366   I 

56 

868680 

4 

40 

I3i320 

32 

29 

774217 

2 

85 

903272    I 

56 

868945 

4 

40 

i3io55 

3i 

3o 

774388 

2 

84 

905179   « 

56 

869209 

4 

40 

130794 

3o 

3i 

9-774558 

2 

84 

9-9o5o85   I 

56 

9-869473 

4 

40 

io.i3o527 

29 
28 

32 

774729 

2 

84 

904992   I 

56 

869737 

4 

40 

i3o263 

33 

774899 

2 

84 

904898   I 

56 

87C001 

4 

40 

1 29999 
129735 

27 

34 

773070 

2 

84 

904804   I 

56 

870265 

4 

40 

26 

35 

775240 

2 

84 

9047 1 1    ' 

56 

870529 

4 

40 

129471 

25 

36 

77^410 

2 

83 

904617   I 

56 

870793 

4 

40 

I2g207 

24 

u 

775580 

2 

83 

904323   I 

56 

871037 

4 

40 

12S943 

23 

773750 

2 

83 

904429   I 

57 

871321 

4 

40 

128679 
128415 

22 

39 

775920 

2 

83 

904335   I 

^ 

871585 

4 

40 

21 

40 

776090 

2 

83 

904241   I 

57 

871849 

4 

39 

I28i5i 

20 

41 

9-776259 

2 

83 

9904147   1 

5: 

9- 8721 1 2 

4 

39 

10-127S88 

19 

42 

776420 

2 

82 

904033   I 

5: 

872376 

4 

39 

127624 

10 

43 

776598 

2 

82 

903959   I 

5- 

872640 

4 

39 

127360 

\l 

44 

776768 

2 

82 

903864   I 

57 

872903 

4 

39 

127097 

43 

776937 

2 

82 

903770    I 

^7 

873167 

4 

39 

1 26833 

i5 

46 

777106 

2 

82 

903676    I 

57 

873430 

4 

39 

126570 

14 

47 

777275 

2 

81 

90358 1  ■  I 

57 

873694 

4 

39 

i263o6 

i3 

48 

777444 

2 

81 

903487  ;  I 

57 

873957 

4 

39 

126043 

12 

49 

777613 

2 

Si 

903392   I 

58 

874220 

4 

39 

125780 

II 

5c 

777781 

2 

81 

903298   I 

58 

874484 

4 

39 

I255i6 

10 

5i 

9-777950 

2 

81 

9 -903203   I 

58 

9 •"74747 

4 

39 

10-125253 

t 

52 

778119 

2 

81 

903 1 08   I 

58 

B75010 

4 

39 

124990 

53 

778287 

2 

80 

9o3oi4   I 

58 

875273 

4 

38 

124727 

I 

54 

778455 

2 

80 

902919   I 

58 

875536 

4 

33 

124464 

55 

778624 

2 

80 

902824   : 

58 

875800 

4 

38 

124200 

5 

56 

778792 

2 

•80 

902729   I 

58 

876063 

4 

38 

123937 

4 

u 

778960 

2 

•  80 

902634   I 

58 

876326 

4 

38 

123674 

3 

779128 

2 

.80 

902539   I 

59 

816589 

4 

38 

123411 

3 

59 

779295 

2 

•79 

902444   I 

59 

876S51 

4 

38 

123140 

I 

60 

779463 

2-79 

902349   I 

59 

877114 

4-38 

I228b6 

0 

Coeine 

D. 

Sine   1   I 

). 

Cotang. 

T). 

Tsiig. 

M. 

(53  DEGREES.) 


SINES  AND  TANGENTS.      (37    DEGR5ESJ 


65 


IS.. 

Sino 

D. 

Cosine 

D. 

Tang. 

D. 

Coiaug. 

60 

0 

9-779463 

a-79 

9-902349 
902253 

,.59 

9'877ii4 

4-38 

10-122886 

I 

779601 

2- 

79 

1.59 

877377 

4 

38 

122623 

u 

3 

77979S 

2- 

79 

902 1 58 

1-59 

877640 

4 

38 

i2236o 

3 

779966 

2- 

79 

902063 

1.59 

877903 

4 

38 

122097 
121835 

57 

4 

7S0133 

2- 

79 

90 1 967 

,.59 

878165 

4 

38 

56 

5 

780300 

2- 

78 

901872 

,.59 

878428 

4 

38 

121572 

55 

6 

780467 

2 

78 

901776 
901681 

1.59 

878691 

4 

38 

i2i3o9  '  54  1 

i 

780634 

2 

78 

1.59 

878953 

4 

37 

121047 

53 

780801 

2 

78 

901585 

1.59 

879216 

4 

3- 

120784 

52 

9 

780968 

2 

78 

901490 

1.59 

879478 

4 

37 

120522 

5i 

10 

781134 

2 

78 

901394 

i.6c 

879741 

4 

37 

120259 

Co 

II 

9-78i3oi 

2 

77 

9-901298 

1-60 

9-88ooo3 

4 

37 

10-119997 

49 

n 

781468 

2 

77 

901202 

I -60 

880265 

4 

37 

119735 

48 

i3 

781634 

2 

77 

901 106 

I -60 

880528 

4 

37 

119472 

47 

U 

781800 

2 

77 

901010 

I -60 

880790 
88io52 

4 

37 

119210 

46 

ID 

781966 

2 

77 

900914 

I -60 

4 

37 

118948 

45 

i6 

782132 

2 

77 

900818 

1-60 

88i3i4 

4 

37 

118686 

44 

11 

782298 

2 

76 

900722 

I -60 

881576 

4 

37 

118424 

43 

i8 

782464 

2 

76 

900626 

1.60 

881839 

4 

37 

I18161 

42 

'9 

782630 

•  2 

76 

900529 

1-60 

88::oi 

4 

37 

117899 

41 

20 

782796 

2 

76 

900433 

1-61 

882363 

4 

36 

117637 

40 

21 

9-782961 

2 

76 

9-900337 

i-6i 

9-882625 

4 

36 

I0-I17375 

39 

22 

783127 

2 

76 

900240 

i-6i 

882887 

4 

36 

117113 

38 

23 

783292 

2 

75 

900144 

1  6j 

883148 

4 

36 

116852 

37 

24 

7834.18 

2 

75 

900047 

i-6i 

883410 

4 

36 

1 16590 

36 

25 

783623 

2 

75 

899951 

i-6i 

883672 

4 

36 

116328 

35 

26 

783788 

2 

75 

899854 

1.61 

883934 

4 

36 

116066 

34 

11 

783953 

2 

75 

899757 

i-6i 

884196 

4 

36 

ii58o4 

33 

7841 18 

2 

75 

899660 

1-61 

884457 

4 

36 

1 1 5543 

32 

29 

784282 

2 

74 

899564 

i-6i 

884719 

4 

36 

ii528i 

3i 

JO 

784447 

2 

74 

899467 

1-62 

884980 

4 

36 

ii5o20 

3o 

3i 

9-784612 

2 

74 

9-899370 

1.62 

9-885242 

4 

36 

10-114758 

20 

28 

32 

784776 

2 

74 

■  -  899373 

1-62 

8855o3 

4 

36 

1 1 4497 

33 

784941 

2 

74 

899176 

1-62 

885765 

4 

36 

114235 

27 

34 

785io5 

2 

74 

890078 

1-62 

886026 

4 

36 

113974 

26 

35 

785269 

2 

73 

898981 

1-62 

886288 

4 

36 

113712 

25 

36 

785433 

2 

73 

898884 

1-62 

886549 

4 

35 

11 3431 

24 

37 

785597 

2 

73 

898787 

1-62 

886810 

4 

35 

Ii3i90 

23 

38 

783761 

2 

73 

898689 

1-62 

887072 

4 

35 

112928 

22 

39 

785925 

2 

73 

898592 

1-62 

887333 

4 

35 

112667 

21 

40 

786089 

2 

73 

898494 

1-63 

887594 

4 

35 

1 1 2406 

20 

41 

9-786252 

2 

72 

9-^98397 

1-63 

9-887855 

4 

35 

10-112145 

'? 

42 

786416 

2 

72 

898299 

1-63 

888116 

4 

35 

1 11884 

18 

43 

786579 

n 

72 

89820a 

1-63 

888377 

4 

35 

111623 

17 

44 

786742 

2 

7^ 

898 lOA 

■  1-63 

888639 

4 

35 

iii36i 

16 

45 

786906 

2 

72 

898006 

63 

888900 

4 

35 

11 1 100 

i5 

46 

787060 

2 

72 

897908 

1-63 

889160 

4 

35 

110841  ;  14  1 

47 

787232 

2 

71 

897810 

1-63 

889421 

4 

35 

110579 

i3 

48 

787395 

2 

7' 

897712 

1-63 

889682 

4 

35 

iio3i8 

12 

49 

787567 

2 

71 

897614 

1-63 

889943 

4 

35 

110057 

II 

DO 

787720 

2 

71 

897516 

1-63 

890204 

4 

34 

109796 

10 

5i 

9.737883 
788045 

2 

71 

9-897418 

1-64 

9-890465 

4 

34 

10-109535 

9 

52 

2 

71 

897320 

1-64 

890725 

4 

.34 

109275 

8 

63 

788208 

2 

71 

897222 

1-64 

890986 

4 

34 

109014 

7 

ti 

788370 

2 

70 

897123 

1-64 

891247 

4 

•34 

108753 

6 

55 

788532 

2 

•  70 

897025 

I -64 

891507 

4 

•  34 

108493 
108232 

5 

56 

7S8694 

2 

•  70 

896926 

1-64 

891768 

4 

•34 

4 

u 

7888D6 

2 

■  70 

896828 

1-64 

892028 

4 

•34 

107972 

3 

789018 

2 

-70 

896729 

1-64 

892289 

4 

•34 

107711 

2 

59 

789180 

2 

•  70 

896631 

1-64 

892549 

4 

•34 

1 0745 1 

I 

6o 

789342 

2-69 

896532 

1-64 

892810 

4-34 

107190 

0  j 

Cosine 

1   D- 

Sine 

D. 

Cotang. 

D. 

Tang.  1  !£. 

(52    DEGREES.) 


36 

(33 

DEGREES.)  A  TiBIE 

OF  LOGARITHMIO 

M. 

Sine 

D. 

Cosine    D 

. 

Tung. 

D. 

Cotang, 

o 

9-789342 

2-69 

9-896532   I- 

64 

9-892810 

4-34 

10-107190 

60 

I 

789304 

2-69 

896433   I  - 

65 

898070 

4-34 

106980 

59 

a 

789665 

2-69 

896335   I  - 

65 

898331 

4-34 

1 06669 

58 

3 

789827 

2-69 

896286   1- 

65 

898391 

4-84 

106409 

ll 

4 

789988 

2 -69 

896187   1- 
896038   1  - 

65 

898831 

4-34 

1 06 1 49 

56 

5 

790149 

2-69 

65 

894111 

4.34 

105889 

55 

6 

790810 

2-68 

893989   1  - 

65 

894371 

4.34 

105629 

54 

7 

790471 

2-68 

895840   1  - 

65 

894682 

4-33 

105868 

53 

8 

790632 

2-68 

895741   1- 

65 

894892 

4-33 

io5io8 

52 

9 

790793 

2-68 

893641   1 

65 

895132 

4-38 

104848 

5i 

10 

790934 

2-68 

895542   I 

65 

895412 

4-33 

104588 

5o 

II 

9-79III5 

2-68 

9.895443   I 

66 

9-895672 

4-33 

10-104828 

49 

12 

791275 

2-67 

895343   1 

66 

895982 

4-88 

104068 

48 

i3 

791436 

2-67 

895244   I 

66 

896192 

4-38 

108808 

47 

14 

791596 

2-67 

895145   I 

66 

896452 

4-38 

103548 

46 

i5 

79'737 

2-67 

893045   I 

66 

896712 

4-38 

108288 

45 

i6 

791917 

2-67 

894945   1 

66 

896971 

4-83 

108029 

44 

n 

792077 

2-67 

894846   1 

66 

897281 

4-88 

102769 

43 

i8 

792237 

2-66 

894746   1 

66 

897491 

4-88 

102309 

42 

'9 

792397 

2-66 

894646   1 

66 

897731 

4-33  ■ 

102249 

41 

20 

792557 

2-66 

894546   I 

66 

898010 

4-33 

10 1 990 

40 

21 

9-792716 
792876 

2-66 

9-894446   1 

67 

9-898270 

4-83 

10-101730 

39 

38 

22 

2-66 

894846   I 

67 

898530 

4-33 

101470 

23 

793o35 

2-66 

894246   I 

67 

898789 

4-83 

I0I2H 

37 

24 

793io5 
793354 

2-65 

894146   1 

67 

899040 

4-82 

100951 

36 

25 

2-65 

894046   1 

67 

899808 

4-82 

100602 
100432 

35 

26 

793514 

2-65 

898946   1 
890046   1 

67 

899368 

4-32 

34 

11 

798673 

2-65 

67 

899827 

4-82 

1 001 73 

33 

793832 

2-65 

898745   I 

67 

900086 

4-82 

099914 

32 

29 

793991 

2-65 

898645   1 

67 

900846 

4-32 

099654 

3i 

1  3o 

794i5o 

2-64 

898544   I 

67 

900605 

4-82 

099895 

3o 

1  3. 

9 -794303 

2-64 

9-898444   I 

68 

9-900864 

4-32 

10-090136 

29 

1  32 

794467 

2-64 

898848   I 

68 

901124 

4-82 

098876 

28 

33 

794626 

2-64 

898248   1 

68 

901888 

4-32 

0986 1 1 

11 

1  34 

794784 

2-64 

898142   1 

68 

901642 

4-32 

098358 

1  35 

794942 

2-64 

898041   1 

68 

901901 

4-32 

098099 

25 

36 

795101 

2-64 

892940   I 

68 

902160 

4-32 

097840 

24 

37 

795259 

2-63 

892889   I 

68 

902419 

4-32 

097381 

23 

38 

795417 

2-63 

892739   I 

68 

902679 

4-82 

097821 

22 

39 

795575 

2-63 

892688   1 

68 

902988 

4-32 

097062 

21 

40 

795733 

2-63 

892536   I 

68 

908197 

4-3i 

096808 

20 

41 

9-795891 

2-63 

9-892435   I 

69 

9-908455 

4-3i 

10-096545 

19 
IS 

42 

796049 

2-63 

892334   1 

6q 

908714 

4-3i 

096286 

43 

796206 

2-63 

892288   1 

69 

908978 

4-81 

006027 

17 

44 

796364 

2-62 

892182   1 

69 

904282 

4-3i 

093768 

16 

45 

796521 

2-62 

892080   I 

69 

904491 

4-3i 

095509 

i5 

46 

796679 

2-62 

891929   1 

69 

904730 

4-3i 

095230 

14 

s 

796836 

2-62 

891827   1 

69 

903008 

4-81 

094992 

i3 

796993 

2-62 

891726   1 

-69 

903267 

4-3i 

094733 

12 

49 

797130 

2-61 

891624   I 

-69 

905526 

4-3i 

094474 

11 

5o 

797307 

2-61 

891523   1 

.70 

905784 

4-81 

094216 

10 

5i 

9.797464 

2-61 

9-891421   1 

•  70 

9.906043 

4-3i 

10.098957 
098698 

9 

53 

797621 

2-61 

891319   1 

-70 

906802 

4-3i 

8 

53 

797777 

2-61 

891217   1 

•  70 

9o656o 

4-3i 

098440 

7 

54 

797934 

2-61 

891115   1 

-70 

906819 

4-3i 

098181 

6 

55 

798091 

2-61 

891018   1 

-70 

907077 

4-3i 

092928 

5 

56 

798247 
798403 

2-61 

89091 1   I 

-70 

907886 

4-31 

092664 

4 

IS 

2-60 

890809   I 

-70 

907594 
907852 

4-3i 

092406 

3 

798560 

2-60 

890707   1 

-70 

4-31 

092  I  48^ 

3 

59 

798716 

2-60 

8qo6o5  [     1 

.70 

908111 

4-3o 

091889 

I 

66 

1 

79887a 

2- 60 

89o5o3    I 

.70 

908869 

4-3o 

091631 

0 

Coeine 

D. 

Sine   1  I 

). 

Cotang. 

D. 

Tanp, 

J\I^ 

(51  m 

:gr] 

SES.) 

SINES   AND  TANGENTS.      (39    DEGREES.) 


57 


M. 

Siue 

D. 

Cosino 

D. 

Tang. 

D. 

Cotaug. 

0 

9.798872 

2 -60 

9 . 890503 

1.70 

9.908369 

4-3o 

ic.09i63i 

60 

I 

799C28 

2 -60 

890400 

1. 71 

908628 

4-3o 

091372 

2 

799184 

2-60 

890298 

1-71 

908886 

4-3o 

OQII14 

3 

799339 

2.59 

890195 

1-71 

909144 

4-3o 

090856 

57 
j6 

4 

799493 

2.59 

800093 
889990 

1.71 

909402 

4-3o 

090598 

5 

799601 

2.59 

1-71 

909660 

4-3o 

090340 

55 

6 

799806  ; 

2.59 

889888 

1. 71 

909918 

4-3o 

090082 

54 

2 

799962 
8001 17 

2.59 

889785 

1. 71 

910177 

4-3o 

089823 

53 

2-59 

889682 

1-71 

910435 

4-3o 

089565 

32 

P 

800272 

2-58 

889579 

1. 71 

910693 
910951 

4-3o 

089307 

5i 
5o 

lo 

800427 

2-58 

889477 

1. 71 

4-3o 

089049 

II 

9.800582 

2-58 

9.889374 

1-72 

9.911209 

4.30 

10.088791 

49 
43 

12 

800737 

2.58 

889271 

1-72 

91 1467 

4-3o 

088533 

IJ 

800892 

2.58 

889168 

1.72 

911724 

4-3o 

088276 

47 

14 

801047 

2.58 

889064 

1-72 

911982 

4-3o 

088018 

46 

i5 

80120I 

2.58 

888961 

1-72 

912240 

4-3o 

087760 

45 

i6 

8oi356 

2.57 

888858 

1-72 

912498 

4-3o 

087502 

44 

1  ..  1 

8oi5ii 

2.57 

888755 

1-72 

912756 

4-Ju 

087244 

43 

i8 

801 665 

2.57 

888651 

1.72 

9i3oi4 

4-29 

086986 

42 

19 

80181Q 
801973 

2.57 

888548 

1.72 

913271 

4-29  , 

086729 

41 

20 

2.57 

888444 

1-73 

913529 

4-29 

086471 

40 

21 

9.802128 

2-57 

9.888341 

1.73 

9.913787 

4-29 

10.086213 

39 

22 

802282 

2-56 

888237 

1.73 

914044 

4-29 

085956 

38 

23 

802436 

2.56 

888134 

1-73 

914302 

4-29 

080698 

V 

24 

802589 
802743 

2.56 

888o3o 

1.73 

914560 

4-29 

085440 

30 

25 

2.56 

887926 

1.73 

914817 

4-29 

o85i83 

35 

26 

802897 

2-56 

887822 

1-73 

9i5oi5 

4-29 

084925 

34 

27 

8o3oDo 

2.56 

887718 

1.73 

915332 

4-29 

084668 

33 

28 

8o32o4 

2.56 

887614 

1-73 

915590 

4-29 

084410 

\^    1 

29 

803357 

2.55 

887510 

1-73 

915847 

4-29 

084153 

\' 

3o 

8o35u 

2.55 

8S7406 

1-74 

916104 

4-29 

o838g6 

30  , 

3i 

9 -803664 

2-55 

9.887302 

1-74 

9-916362 

4-29 

10.083638 

29 

32 

803817 

2-55 

887198 

1-74 

916619 

4-29 

o8338i 

28 

33 

803970 

2-55 

887093 

1-74 

916877 

4.29 

oS3i23 

27 

34 

804123 

2-55 

8S6989 

1-74 

917134 

4-29 

082866 

26 

35 

804276 

2-54 

886885 

1-74 

917391 

4.29 

082609 

25 

36 

804428 

2.54 

886780 

1-74 

917648 

4-29 

082352 

24 

37 

804581 

2-54 

886676 

1-74 

917905 
918163 

4-20 

4-28 

082095 

23 

3^ 

804734 
804886 

2.54 

886571 

1-74 

081837 

22 

4o 

2-54 

886466 

1-74 

918420 

4.28 

o8i58o 

21 

8o5o39 

2.54 

886362 

1.75 

918677 

4.28 

o8i323 

20 

4i 

9-805191 

2.54 

9.886257 

1.75 

9 •918934 

4.28 

10.081066 

18 

42 

805343 

2.53 

886i52 

1.75 

919191 

4-28 

080809 

43 

805495 

2.53 

886047 

1.75 

919448 

4.28 

o8o552 

\l 

44 

8o5647 

2.53 

885942 

1.75 

919705 

4.28 

080295 

45 

805799 

2.53 

885837 

.1.75 

919962 

4-28 

o8oo38 

i5 

46 

805931 

2-53 

885732 

1.75 

920219 

4-28 

079781 

14 

47 

806 I o3 

2-53 

885627 

1.75 

920476 

4-28 

079524 

i3 

48 

806254 

2.53 

883522 

1-75 

920733 

4.28 

079267 

13 

49 

806406 

2.52 

885416 

1.75 

920990 

4-28 

079010 

II 

5o 

806557 

2-52 

885311 

1.76 

921247 

4-28 

078753 

10 

5i 

9-806709 

2.52 

9-885205 

1.76 

9.921503 

4-28 

10.078497 

I 

52 

806860 

2-52 

885100 

1.76 

921760 

4-28 

078240 

53 

80701 1 

2-52 

884994 
884889 
884783 

1-76 

922017 

4.28 

0779-^3 

I 
5 

54 

807163 

2.52 

1-76 

922274 

4.28 

077726 

35 

807314 

2-52 

1.76 

922530 

4-28 

077470 

56 

807465 

2.5i 

884677 

1-76 

9227S7 

4-28 

077213 

4 
3 

5i 

807615 

2.5l 

884572 

1.76 

923644 

4-28 

076956 

807766 

2.5l 

884466 

1.76 

923300 

4-28 

076700 

3 

5g 

807917 
806067 

2-51 

884360 

..76 

923557 

4-27 

076443 

i 

60 

2.5l 

884254 

1-77 

9238i3 

4.27 

076187 

0 

I 

1  Cosine 

1   D. 

Sine 

1   D. 

1  Cotan?. 

D. 

Tang. 

M. 

(5C 

1  DEOU 

KES.') 

38 

(40 

DEGREES.)   A  TABLE 

OF  LOGARITHMIC 

M. 

n 

Sine 

D. 

Cosine    I 

). 

Tail-,'. 

D. 

Cotang. 

9-808067 

2-5l 

9.884254   1 

77 

9-923813 

4-27 

10  076187 

60 

I 

808218 

2 

5i 

884148   I 

77 

924070 

4 

•27 

076930 

S 

3 

8o8368 

2 

5i 

884042   1 

77 

^24327 

4 

27 

076673 

3 

8o85i9 

2 

5o 

883936    1 

77 

924583 

4 

27 

076417 

57 

4 

808669 

2 

5o 

883829    I 

77 

924840 

4 

27 

076160 

56 

5 

808819 

2 

5o 

883723    I 

77 

923096 

4 

27 

074004 

55 

6 

808969 

2 

5o 

883617   I 

77 

9253D2 

4 

27 

074643 

54 

7 

8091 19 

2 

5o 

883510    I 

77 

923609 

4 

27 

074391 

53 

8 

809269 

2 

5o 

883404    I 

77 

925865 

4 

27 

074135 

52 

9 

809419 

2 

49 

883297    I 

78 

926122 

4 

27 

073878 

5i 

10 

809669 

2 

49 

883191    1 

78 

926378 

4 

27 

073621 

50 

II 

9- 80971 8 
809868 

2 

49 

9 -883084   I 

78 

9-926634 

4 

27 

10-073366 

^9. 

13 

2 

49 

882077   I 
882871    I 

78 

926890 

4 

27 

073110 

4S 

i3 

&10017 

2 

49 

78 

927147 

4 

27 

072853 

47 

14 

810167 

2 

40 

882764   I 

78 

927403 

4 

27 

072697 

46 

i5 

8io3i6 

2 

48 

882657   1 

78 

927639 

4 

27 

072341 

45 

i6 

810465 

2 

48 

882 55o   I 

78 

927915 

4 

27 

072085 

44 

\l 

810614 

2 

48 

882443   I 

78 

928171 

4 

27 

071829 
071673 

43 

810763 

2 

48 

882336   I 

79 

928427 

4 

27 

42 

19 

810913 

2 

48 

882229   I 

79 

928683 

4 

27 

071317 

41 

20 

811061 

2 

48 

882121    I 

79 

928940 

4 

27 

071060 

40 

21 

9-811210 

2 

48 

9-882014   1 

79 

9-929196 
929452 

4 

27 

10-070804 

3q 

22 

8ii358 

2 

47 

881907   I 

79 

4 

27 

070648 

38 

23 

8ii5o7 

2 

47 

881799   I 

79 

929708 

4 

27 

070292 

37 

24 

8ii655 

2 

47 

881692   I 

79 

929964 

4 

26 

070036 

36 

25 

P.I  1804 

2 

47 

881 584   I 

79 

9I0220 

4 

26 

069780 

35 

26 

81 1952 

2 

47 

881477    I 

79 

930475 

4 

26 

069625 

34 

11 

812100 

2 

47 

881369   ' 

79 

930731 

4 

26 

069269 
0600 1 J 

33 

812248 

2 

47 

8S1261    I 

80 

930987 

4 

26 

32 

29 

812396 

2 

46 

881153    1 

80 

931243 

4 

26 

06S767 

3i 

3o 

812544 

2 

46 

881046   I 

80 

931499 

4 

26 

068601 

3o 

3i 

.5-812692 

2 

46 

9-880938   I 
88o83o   I 

80 

9-931755 

4 

26 

10-068245 

29 

32 

812840 

2 

46 

80 

932010 

4 

26 

067990 

28 

33 

812988 

2 

46 

880722   I 

80 

932266 

4 

26 

067704 

27 

34 

8i3i35 

2 

46 

880613   1 

80 

932522 

4 

26 

067478 

26 

35 

Si3283 

2 

46 

88o5o5   I 

So 

932778 

4 

26 

067222 

25 

36 

8i343o 

2 

45 

880397   I 

80 

933o33 

4 

26 

066967 

24 

3? 
3S 

813578 

2 

45 

880289   I 

81 

933289 

4 

26 

066711 

23 

813725 

2 

45 

880180   I 

81 

o33543 

4 

26 

066455 

22 

3q 

813872 

2 

45 

880072   I 

81 

933800 

4 

26 

066200 

21 

40 

814019 

2 

45 

879963   I 

81 

934066 

4 

26 

066944 

20 

41 

9-814166 

2 

45 

t.- 870855   I 

81 

9 -9343 1 1 

4 

26 

10- 066680 

19 

42 

8i43i3 

2 

45 

879746   I 

81 

934667 

4 

26 

066433 

18 

43 

814460 

2 

44 

8796:;7    I 

81 

934823 

4 

26 

066177 

17 

44 

814607 

2 

44 

879620   I 

81 

935078 

4 

26 

064922 

16 

45 

814753 

2 

44 

879420   I 

Si 

935333 

4 

26 

064667 

i5 

46 

814900 

2 

44 

879311    I 

81 

936689 

4 

26 

06441 1 

14 

8 

81 5046 

2 

44 

87920:   I 

82 

935844 

4 

26 

064166 

i3 

815193 

2 

44 

879093    I 
878984  '  : 

82 

936100 

4 

26 

063900 

12 

49 

8i533g 

2 

44 

82 

936355 

4 

26 

063645 

11 

5o 

8 15485 

2 

43 

878875   I 

82 

936610 

4 

26 

063390 

10 

5i 

9-8i563i 

2 

43 

9-878766   I 

82 

1-936866 

4 

26 

lo-o63i34 

0 

5j 

815778 

2 

43 

878656   I 

82 

931121 

4 

23 

062879 

S 

53 

815924 

2 

43 

878547   I 

82 

937376 

4 

26 

062624 

7 

54 

816069 

2 

43 

878433   I 

82 

937632 

4 

25 

062363 

6 

55 

816215 

2 

43 

878328   I 

82 

937887 

4 

25 

062113 

5 

56 

8i636i 

2 

43 

87S219   I 

S3 

938141 

4 

26 

061868 

4 

§ 

8i65o7 

2 

42 

878109   I 

83 

933398 

4 

25 

061602 

3 

8i6652 

2 

42 

877999   I 

83 

93863S 

4 

25 

06 1 347 

2 

59 

8i6-q3 

2 

42 

877800   I 

83 

93S908 

i 

2'. 

06100'' 
o6o83t 

I 

66 

816943 

2-42 

877780   I 

83 

939163 

4 -a- 

0 

Coaine 

D. 

Sine   1  r 

). 

Cotang. 

I>._^ 

Tang. 

■Nr. 

(49  DEGREES.) 


SINES  AND  TANGENTS. 

1.41  DEGREES.) 

^ 

M. 

0 

Sine 

D.   1 

Cosiac 

D. 

Tung. 

D. 

Cotang. 

60 

9.816943 

8170S8 

1 

2-42 

9.377780 

•  83 

9.939163 

4-25 

10-060837 

I 

2-42 

877670 

1-83 

939418 

4-25 

o6o582 

59 
58 

2 

817233 

2-42 

877560 

1-83 

939673 

4-25 

o6o327 

3 

817379 

2-42 

8774':o 

1-83 

939928 

4-i5 

060072 

57 

4 

817524 

2-41 

877340 

1-83 

940183 

4-25 

059817 

56 

5 

817668 

2-41 

877230 

1-84 

940438 

4-25 

o5g562 

55 

6 

817813 

2-41 

877120 

1-84 

940694 

4-25 

0D9306 

&4 

817953 

2-4i 

877010 

1-84 

940949 

4-25 

o5oo5i 

53 

8i8io3 

2-41 

876899 

1-84 

941204 

4-25 

058796 

52 

9 

818247 

2-41 

876789 

1-84 

941458 

4-25 

058542 

5i 

xO 

818392 

2-41 

876678 

1-84 

941714 

4-25 

058286 

5o 

II 

9. 8 18536 

2-40 

9.876568 

1-84 

g. 941968 

4-25 

io-o58o32 

4q 

I3 

818681 

2-40 

876457 

1-84 

942223 

4  23 

057777 

48 

i3 

818825 

2-40 

876347 

1-84 

942478 

4-35 

057522 

47 

i4 

818969 
819113 

2 -40 

876236 

1-85 

942733 

4-25 

057267 

46 

i5 

2-4° 

876125 

1-85 

942988 

4-25 

057012 

45 

i6 

819257 

2-40 

876014 

1-85 

943243 

4-25 

050757 

44 

17 

819401 

2-40 

875904 

1-85 

943498 

4-25 

056302 

43 

i8 

819545 

2-39 

875793 

1.85 

943752 

4-25 

056248 

42 

19 

8196B9 

2-39 

875682 

1-85 

944007 

4-25 

055993 

41 

20 

819832 

2-39 

875571 

1.85 

944262 

4.35 

055738 

40 

21 

9-819976 

2-39 

9.875459 

1.85 

9.944517 

4-25 

10-055483 

39 

22 

820120 

2-39 

875348 

1-85 

944771 

4-24 

055229 

38 

23 

820263 

2-39 

875237 

1-85 

945026 

4-24 

054974 

37 

24 

820406 

2-39 

875126 

1-86 

945281 

4-24 

054719 

36 

13 

82q55o 

2-38 

875014 

1-86 

945535 

4-24 

054465 

35 

26 

820603 
82o8,)6 

2-38 

874903 

1-86 

940790 

4-24 

054210 

34 

27 

2-38 

87479' 

1-86 

946045 

4-24 

053955 

33 

28 

820979 

2-38 

874680 

1-86 

9462(^9 

4-24 

0537CI 

32 

29 

82 1122 

2-38 

874568 

1-86 

9465D4 

4-24 

053446   31 

3o 

821265 

2-38 

874436 

1-86 

946808 

4-24 

053192   3o 

3i 

9-821407 

2-38 

9-874344 

1-86 

g. 947063 

4-24 

io-o52(;3t 

20 

32 

82i55o 

2-33 

874232 

1-87 

947318 

4-24 

052682 

28 

33 

821693 

2.37 

874121 

1-87 

947372 

4-24 

052428 

27 

34 

821835 

2-37 

874009 

1-87 

947826 

4-24 

052174 

26 

35 

821977 

2-37 

873896 

1-87 

94808 1 

4-24 

051919 

25 

36 

822120 

2-37 

873784 

1-87 

948336 

4-24 

o5i664 

24 

37 

822262 

2-37 

873672 

1-87 

948590 

4-24 

o5i4io 

23 

38 

822404 

2-37 

873560 

1-87 

948844 

4-24 

o5ii56 

22 

39 

822546 

2-37 

87344S 

1-87 

949090 
949333 

4-24 

050901 

21 

40 

822688 

2-36 

873335 

1-87 

4-24 

050647 

20 

41 

9.822830 

2-36 

9-873223 

1.88 

9.949607 

4-24 

10-050393 

o5oi38 

19 
l8 

42 

822972 

2.36 

8731 10 

g4gS62 

4-24 

43 

823114 

2-36 

872998 

1.83 

950116 

4-24 

049884 

17 

44 

823255 

>.36 

872885 

1.88 

950370 

4-24 

049630 

16 

43 

823397 

2-36 

1   872772 

1-83 

950625 

4-24 

049375 

i5 

46 

'■       82353g 

2-36 

872659 

.  1.88 

g5o879 

4-24 

04g 1 2 1 

14 

47 

.   82  3686 

2-35 

872547 

■  1.88 

95ii33 

4-24 

04S867 

i3 

48 

823821 

2-35 

872434 

1.83 

95i3S8 

4-24 

048612 

12 

49 

823963 

2-35 

872321 

1.83 

951642 

4-24 

048358 

11 

5o 

824104 

2-35 

872208 

1-83 

951896 

4-24 

048 1 04 

10 

5i 

9-824245 

2-35 

1  9-872095 

1-89 

g.g52i5o 

4-24 

io.o47H5o 

9 

52 

824386 

2-35 

1   871981 
87186S 

1-89 

952405 

4-24 

047395 

8 

53 

824527 
824668 

2-35 

1.89 

95265q 
952gi3 

4-24 

047341 

7 

54 

2-34 

871755 

1-89 

4-24 

047087 

6 

55 

824808 

2-34 

871641 

1.89 

953167 

4-23 

046833 

5 

56 

,   824949 

2-34 

87028 

1.89 

953421 

4-23 

046579 

4 

u 

'   825090 
825230 

2-34 

871414 

1-89 

953675 

4-23 

046325 

3 

2-34 

871301 

i.8q 

953929 

4-23 

046071  ,  2  1 

59 

825371 

2-34 

871187 

1-89 

954183 

4-23 

045817 

I 

60 

825511 

2-34 

871073 

1-90 

954437 

4-23 

045563 

c 

Cosine 

D. 

Sine 

D. 

Cotang. 

U 

T-uwf. 

M. 

(48    USGRKBS.) 


60 


(42   DEGREES.)     A  TABLE   OF   LOGARITHMIO 


M. 

Sine  , 

D.   1 

Cosine    E 

>. 

Tang. 

D. 

Cotang. 

0 

9-825511 

2-34 

9-871073   1 

90 

9-954437 

4.23  1 

ic  045563 

fto 

1 

825651 

2- 

33 

870960   1 
870846   1 

90 

954691 

4- 

23 

o453oq 

U 

2 

823791 

2- 

33 

90 

954945 

4- 

23 

045o55 

3 

823931 

2- 

33 

870732   1 

90 

955200 

4 

23 

044800  1  57 

4 

826071 

2- 

33 

870618   1 

90 

955454 

4- 

23 

044546  i  56 

5 

826211 

2- 

33 

870504   I  • 

90 

955707 

4- 

2? 

044293  j  55 

6 

826351 

2- 

33 

870390   1- 

90 

955961 

4 

23 

o44o3g  ]  54 

7 

826491 

2- 

33 

870276   I 

90 

956215 

4- 

23 

043783  ;  53 

8 

826631 

2- 

33 

870161    I 

90 

956469 
956723 

4 

23 

043531  I  53 

9 

826-70 

2- 

32 

870047   1 

91 

4 

23 

043277 

5i 

10 

826910 

2- 

32 

869933   I  - 

91 

956977 

4 

23 

o43o23 

5o 

II 

9-827049 

2- 

32 

9-869818   I- 

9' 

9-957231 

4 

23 

10.042769 
o425i5 

40 

13 

827189 

2- 

32 

809704   1 

9' 

957485 

4 

23 

48 

!3 

827328 

2- 

32 

869589   1 

9' 

937739 

4 

23 

042261 

47 

\S 

827467 

2- 

32 

869474   I 

9' 

957993 

4 

23 

042007 

46 

i5 

827606 

2- 

32 

869360   1 

9' 

938246 

4 

23 

041754 

45 

i6 

827745 

2- 

32 

869245   I 

91 

958500 

4 

23 

o4 I 5oo 

44 

17 

821884 

2- 

3i 

869130   I 

91 

95S754 

4 

23 

041246 

43 

i8 

828023 

2- 

3i 

869015   1 

92 

959008 

4 

23 

040992 

42 

19 

828162 

2- 

3i 

868900   1 

92 

959262 

4 

23 

040738 

41 

20 

828301 

2- 

3i 

868785   1 

92 

959516 

4 

23 

040484 

40 

21 

9 -828430 

2- 

3i 

9-868670   I 

92 

9-959769 
960023 

4 

23 

io-o4o23i 

^§ 

22 

82857S 

2- 

3i 

868555   1 

92 

4 

23 

039977 

38 

23 

828716 

2- 

3i 

868440   1 

92 

960277 

4 

23 

039723 

37 

34 

828855 

2- 

3o 

868324   I 

92 

96053 1 

4 

23 

0394 1>9 

36 

25 

828903 
829131 

2- 

3o 

868209   I 
868093   I 

92 

960784 

4 

23 

039216 

35 

26 

2 

3o 

92 

961038 

4 

23 

038962 

34 

S 

829269 

2 

3o 

867978   I 
867862   1 

93 

961291 

4 

23 

038709 

33 

829407 
829545 

2 

3o 

93 

961545 

4 

23 

038455 

32 

29 

2 

3o 

867747   I 

93 

961799 

4 

23 

o382oi 

3i 

3o 

829683 

3 

3o 

867631   1 

93 

962032 

4 

23 

037948 

3o 

3i 

9.829821 

2 

29 

9-867515   I 

93 

9-962306 

4 

23 

10-037694 

29 

32 

829959 

2 

29 

86^^  ; 

93 

962360 

4 

23 

037440 

28 

33 

83oo97 
830234 

2 

29 

93 

962813 

4 

23 

037187 

27 

34 

2 

29 

867167  I 

93 

963067 

4 

23 

036933 

26 

35 

83o372 

2 

29 

867051   1 

93 

963320 

4 

23 

o3668o 

25 

36 

83o5og 

2 

29 

866935   I 

866819   1 

866703   1 

94 

963574 

4 

23 

036426 

24 

37 

83o646 

2 

29 

94 

963827 

4 

'I 

o36i73 

23 

38 

830784 

2 

29 

94 

964081 

4 

-23 

035919 

22 

39 
40 

830921 

2 

28 

866586   1 

94 

964335 

4 

23 

035665 

21 

83io58 

2 

28 

866470   I 

94 

964588 

4 

22 

035412 

20 

41 

9-831195 

2 

28 

9-866353   1 

94 

9-964842 

4 

22 

io-o35i58 

11 

42 

83i332 

2 

23 

866237   I 

94 

965095 

4 

22 

o349o5 

43 

831469 

2 

28 

866120   I 

94 

965349 

4 

22 

o3465i 

17 

44 

83 1606 

2 

28 

866004   I 

95 

965602 

4 

22 

034398 

16 

45 

831742 
831879 
83201 5 

2 

28 

865887   1 

95 

965855 

4 

22 

034145 

i5 

46 

2 

28 

865770   1 

95 

966105 

4 

22 

033891 
033638 

14 

47 

2 

27 

865653   1 

93 

966362 

4 

22 

i3 

48 

832i52 

2 

27 

865536   I 

95 

966616 

4 

22 

033384 

12 

49 

832288 

2 

27 

865419   I 

95 

966869 

4 

2  2 

o33i3i 

II 

50 

832425 

2 

27 

865302   1 

95 

967123 

4 

22 

032877 

10 

5i 

9-832561 

2 

27 

9-865185   I 

95 

9-967376 

4 

22 

10-032624 

t 

52 

832697 

2 

27 

865o68   I 

95 

967629 

4 

22 

032371 

53 

832833 

2 

27 

864950   I 

95 

967883 
968i36 

4 

22 

032117 

7 

54 

832969 
833I0D 

2 

26 

864833   1 

96 

4 

22 

o3i864 

6 

55 

2 

-26 

864716   1 

96 

968389 

4 

22 

o3i6ii 

5 

56 

833241 

2 

-26 

864598   1 

.96 

968643 

4 

22 

o3i357 

4 

s 

833377 

2 

-26 

8644S1   1 

-96 

968896 

4 

22 

o3iio4 

3 

833512 

2 

-26 

864363   1 

96 

969149 

4 

22 

o3o85i 

a 

59 

833648 

2 

-26 

864245   1 

96 

969403 

4 

■22 

o3o597 

I 

60 

833793 

2-26 

864127   I 

-96 

969656 

4-22 

o3o344 

0 

Cosine 

D. 

Sine   '   r 

). 

Cotano;. 

D, 

Tang. 

-M^ 

(47   DKGREES.) 


BINES  AND   TANGENTS.      (i3    DEGREES.; 


61 


M. 

Sine 

D. 

Cosine    D 

. 

Tang.  1 

D.   1 

Cotung. 

60 

o 

9-833783 

2-26 

9-864127   1- 

96 

9  969656 

4.22 

ID- 080844 

I 

833919 

2-25 

864010   1- 

96 

96990Q 

4.22 

080091 

029888 

5^ 

a 

834o54 

2-25 

868S92    I • 

97 

970162 

4.22 

3 

834189 

2-25 

868774   I  • 

97 

970416 

4-22 

029384 

ll 

4 

83432D 

2-25 

868656   I 

97 

970669 

4-22 

029881 

5 

834460 

2-25 

863538   I  - 

97 

970922 

4-2'^ 

029078 

028825 

55 

6 

8345q5 
834730 

2-25 

868419   I 

97 

971173 

4-22 

54 

I 

2-25 

868801    I  - 

97 

971429 

4-22 

028371 

53 

834865 

2-25 

868188   1- 

97 

971682 

4-22 

028818 

52 

9 

834999 
835i34 

2-24 

868064   1  • 

97 

971935 

4-22 

02806'^ 

5i 

10 

2-24 

862946   I  - 

98 

972188 

4-22 

027812 

5o 

II 

9-835269 

2-24 

9-862827   I 

98 

9-972441 

4-22 

10-027559 

49 

la 

8354o3 

2-24 

862709   I 

98 

972694 

4-22 

027806 

48 

i3 

835538 

2-24 

862590   1- 

98 

972948 

4.22 

027032 

47 

14 

835672 

2-24 

862471   I 

98 

978201 

4.22 

026799 

46 

i5 

835807 

2-24 

862853   I 

98 

973454 

4-22 

026546 

45 

i6 

835941 

2-24 

8^2284   I 

98 

978707 

4-22 

026298 

44 

\l 

886075 

2-23 

8621 i5   I 

98 

978960 

4-22 

026040 

48 

836209 
836343 

2-23 

861996   I 

98 

974218 

4-22 

025787 

42 

19 

2-23 

861877   I 

98 

974466 

4-22 

025534 

41 

20 

836477 

2-23 

861758   1 

99 

974719 

4.22 

025281 

40 

21 

9-83661 1 

2-23 

9.861688   I 

99 

9-974973 

4.22 

10.025027 

3o 

32 

836745 

2-28 

86i5i9   I 

99 

975226 

4-22 

024774 

38 

23 

836878 

2-23 

861400   I 

99 

975479 

4-22 

024321 

37 

24 

837012 

2-22 

861280   I 

99 

975782 

4-22 

024268 

36 

25 

837146 

2-22 

861161   1 

99 

973985 

4-22 

024015 

85 

26 

837279 

2-22 

861041   J 

99 

976288 

4-22 

0.28762 

84 

11 

837412 

2-22 

860922   I 

99 

976491 

4-22 

028509 

38 

837546 

2-22 

860802   I 

99 

976744 

4-22 

028256 

32 

29 

837679 

2-22 

860682   2 

00 

976997 
977250 

4.22 

028008 

3i 

3o 

837812 

2-22 

86o562   2 

00 

4.22 

022750 

3o 

3i 

9.837945 

2-22 

9.860442   2 

00 

9-977508 

4.22 

10.022497 

29 

32 

838078 

2-21 

860822   2 

00 

977756 

4.22 

022244 

28 

33 

8382  u 

2-21 

860202   2 

00 

978009 

4-22 

021991 

77 

34 

838344 

2-21 

8600S2   2 

00 

978262 

4-22 

021788 

26 

35 

838477 

2-2! 

839962   2 

00 

9785i5 

4-32 

021485 

25 

36 

838610 

2-21 

859842   2 

00 

978768 

4-22 

021282 

24 

il 

838742 

2-21 

859721   2 

01 

979021 

4-22 

020979 

28 

838875 

2-21 

859601   2 

01 

979274 

4-22 

020726 

22 

39 

889007 

2-21 

859480   2 

01 

979527 

4-22 

020478 

21 

40 

889140 

2-20 

859860   2 

01 

979780 

4-22 

020220 

20 

41 

9-889272 

2-20 

9-859289   2 

01 

9-980083 

4.22 

10.019967 

19 

42 

839404 

2-20 

859119   2 

01 

9802S6 

4.22 

019714 

18 

43 

8S9536 

2-20 

858998   2 

01 

980538 

4.22 

019462 

\l 

44 

8396'.8 

2-20 

858877   2 

01 

9S0791 

4.21 

oig209 

45 

889800 

2-20 

858756   2 

02 

98 1 044 

4.21 

018936 

i5 

46 

889982 

2-2C 

85S685   2 

02 

981297 

4-21 

018708 

14 

a 

840064 

2-19 

85S5i4   2 

02 

981550 

4-21 

018430 

i3 

840196 

2-19 

858398   2 

-02 

981808 

4-21 

018197 

12 

49 

840828 

2-19 

808272   2 

-02 

982056 

4-21 

017944 

n 

5o 

840459 

2-19 

858i5i   2 

-02 

982800 

4-21 

017691 

10 

5i 

9-840591 

2-19 

9.838029   2 

-02 

9.982502 

4-21 

10.017488 

0 

5a 

840722 

2-19 

S57908   2 

■  02 

982814 

4-2! 

017186 

if 

53 

840854 

2-19 

1   857786   2 

-02 

988067 

4-21 

016988 

1 

54 

840985 

857665   2 

-o3 

983820 

4-21 

016680 

6 

55 

841 1 16 

'   2-18 

857543   2 

-o3 

983578 

4-21 

016427 

5 

56 

841247 
841879 

!  2-18 

857422   2 

-o3 

988826 

4.21 

016174 

4 

u 

2-18 

857800   2 

-o3 

984079 

4-21 

015921 

3 

84009 

2-18 

857178   2 

-o3 

984881 

4.21 

015669  ;  a 

59 

841640 

2-l8 

857056   2 

-o3 

9=./ 584 

4-21 

013416 

I 

6o 

841771 

'   2-l8 

1 

856984   2 

•  o3 

984881 

4-21 

oi5i63 

0 

Cosine 

1   D. 

Sine     1 

\. 

Cfitane 

D. 

Tang. 

]m._ 

(46    DEGREES.) 


62 


(44    DEGREES.)     A   TABLE   OF   LOGARITHMIC 


u. 

Sine 

D. 

Cosino 

D. 

Tang. 

D. 

Cotang. 

o 

9-841771 

2.18 

9-856934 

2-o3 

9-984837 

4.21 

1 0 . 0 1 5 1 63 

60 

I 

841902 

2 

18 

856812 

2 

o3 

085090 

4 

21 

014910 

59 

3 

842033 

2 

18 

806690 

2 

04 

985343 

4 

21 

014657 

5§ 

3 

842163 

2 

'7 

856568 

2 

04 

985596 

4 

21 

014404 

11 

4 

842294 

2 

"7 

856446 

2 

04 

985848 

4 

21 

oi4i52 

5 

842424 

2 

17 

856323 

2 

04 

986101 

4 

21 

013899 

5f  , 

6 

842555 

2 

17 

{i562oi 

2 

04 

986354 

4 

21 

013646 

54 

I 

8426S5 

2 

17 

856078 

2 

04 

986607 

4 

21 

013393 

53 

842815 

2 

'7 

855956 

2 

04 

986860 

4 

21 

oi3i4o 

52 

9 

842946 

2 

17 

855833 

2 

04 

987112 

4 

21 

012888 

5i 

10 

843076 

2 

17 

855711 

2 

o5 

987365 

4 

21 

012635 

5o 

II 

9.843206 

2 

16 

9-855588 

2 

o5 

9-987618 

4 

21 

I0-0I23S2 

4o 

12 

843336 

2 

16 

855465 

2 

o5 

987871 

4 

21 

012129 

48 

i3 

843466 

2 

16 

855342 

2 

c5 

988123 

4 

21 

011877 

47 

14 

843595 

2 

16 

855219 

2 

o5 

988376 

4 

21 

011624 

46 

i5 

843725 

2 

16 

855096 

2 

o5 

988629 

4 

21 

011371 

45 

i6 

843S55 

2 

16 

854973 

2 

o5 

98S882 

4 

2  I 

on  118 

44 

\l 

843984 

2 

16 

854850 

2 

o5 

989134 

4 

21 

0 1 0866 

43 

8441 14 

2 

i5 

854727 

2 

06 

989387 

4 

21 

oio6i3 

42 

J9 

844243 

2 

i5 

854603 

2 

06 

989640 

4 

21 

oio36o 

41 

20 

844372 

2 

i5 

854480 

2 

06 

989893 

4 

21 

010107 

40 

21 

9-844502 

2 

i5 

9-854356 

2 

06 

9-990145 

4 

21 

10-009855 

U 

22 

844631 

2 

i5 

854233 

2 

06 

9903^8 

4 

21 

009602 

23 

844760 

2 

i5 

854109 

2 

06 

99063 1 

4 

21 

009349 

u 

24 

844889 

2 

i5 

8539S6 

2 

06 

990903 

4 

21 

009097 
008844 

25 

845018 

2 

i5 

853862 

2 

06 

99 11 56 

4 

21 

35 

26 

845147 

2 

i5 

853738 

2 

06 

991409 

4 

21 

008591 

34 

u 

845276 

2 

14 

853614 

2 

07 

991662 

4 

21 

008338 

33 

845405 

2 

14 

853490 

2 

07 

991914 

4 

21 

008086 

32 

29 

845533 

2 

14 

853366 

2 

07 

992167 

4 

21 

007833 

3i 

3o 

845662 

2 

14 

853242 

2 

07 

992420 

4 

21 

007580 

3o 

3i 

9.845790 

2 

14 

9-853ii8 

2 

07 

9-992672 

4 

21 

10-007328 

;i 

32 

845919 

2 

14 

852994 

2 

07 

992925 

4 

21 

007073 

33 

846047 

2 

14 

852869 

2 

07 

993178 

4 

21 

006822 

27 

34 

846175 

2 

14 

85274D 

2 

07 

993430 

i 

21 

006570 

26 

35 

846304 

2 

14 

852620 

2 

07 

993683 

4 

21 

oo63 1 7 

25 

36 

846432 

2 

i3 

852496 

2 

08 

993936 

4 

21 

006064 

24 

S 

846560 

2 

i3 

852371 

2 

08 

994189 

4 

21 

oo58ii 

23 

846688 

2 

i3 

852247 

2 

08 

994441 

4 

21 

oo555g 

22 

39 

84681b 

2 

i3 

852  122 

2 

08 

994694 

4 

21 

oo53o6 

21 

4o 

846944 

2 

,  1 
1  J 

851997 

2 

08 

994947 

4 

21 

oo5o53 

20 

41 

9.847071 

2 

i3 

9-851872 

2 

oS 

9-995199 

4 

21 

10-004801 

10 
18 

42 

847199 

2 

i3 

851747 

2 

08 

995432 

4 

21 

004548 

43 

847327 

2 

i3 

85i622 

2 

08 

995705 

4 

21 

004295 

17 

44 

847454 

2 

12 

85 1497 

2 

09 

995957 

4 

21 

004043 

16 

45 

847582 

2 

12 

85 1372 

2 

09 

996210 

4 

21 

003790 

i5 

46 

847709 

2 

12 

851246 

2 

09 

996463 

4 

21 

003537 

14 

47 

847836 

2 

12 

851121 

2 

09 

996715 

4 

21 

oo32S5 

i3 

48 

847964 

2 

12 

850996 
850870 

2 

09 

996968 

4 

21 

oo3o32 

12 

49 

848091 

2 

12 

2 

09 

997221 

4 

21 

002779 

II 

5o 

848218 

2 

12 

b5o745 

2 

09 

997473 

4 

21 

002527 

10 

5i 

9.848345 

2 

12 

9-85o6i9 
85o493 

2 

09 

9.997726 

4 

21 

10-002274 

9 

52 

848472 

2 

1 1 

2 

10 

997979 

4 

21 

002021 

8 

53 

848599 

2 

11 

85o368 

2 

10 

998231 

4 

21 

001769 

I 

54 

848726 

2 

II 

850242 

2 

10 

998484 

4 

21 

ooi5i6 

;.5 

848852 

2 

1 1 

85oii6 

2 

10 

998737 

4 

21 

001263 

5 

56 

848979 

2 

11 

849990 
849864 

2 

10 

998989 

4 

21 

00101 1 

4 

57 

849106 

2 

11 

2 

10 

999242 

4 

21 

000758 

3 

58 

849232 

2 

II 

849738 

2 

10 

999493 

4 

21 

ooo5o5 

2 

59 

849359 

2 

11 

849611 

2 

10 

999748 

4 

21 

O00253 

I 

60 

849483 

211 

849485 

2-10 

10.000000 

4.21 

1 0  -  000000 

0 

Cosine 

D. 

Sino 

D. 

C'otangr. 

D. 

Tun?. 

M. 

(45  DEGREES.) 


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